Retina, 48 47 bytes

i`(?<=([aeiou]).*?[b-z-[eiou]])(?![aeiou])
$l$1

Try it online! Explanation: The lookahead searches for a point not followed by a vowel, while the lookbehind searches for an immediately preceding consonant and a previous vowel, which is then inserted in lower case. Edit: Saved 1 byte by using character class subtraction.

Yabasic, 180 bytes

A full program which takes input from STDIN and outputs to STDOUT

Line Input""s$
x$="AEIOUaeiou"
For i=1To Len(s$)
c$=Mid$(s$,i,1)
?c$;
If InStr(x$,c$)Then
v$=c$
Else
a=Asc(Upper$(c$))
If a>64And a<91And!InStr(x$,Mid$(s$,i+1,1))Then?v$;Fi
Fi
Next

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Red, 276 bytes

func[s][v: charset t:"AEIOUaeiou"c: charset 
u:"BCDFGHJKLMNPQRSTVWXYZbcdfghjklmnpqrstvwxyz"b:
parse s[collect[any keep[thru c opt v]keep thru end]]p:""foreach
c b[either find t e: last c: to-string c[p: e][parse c[any[copy p v
| skip]]if find u e[append c lowercase p]]prin c]]

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Readable:

f: func [ s ] [
   v: charset t: "AEIOUaeiou"
   c: charset u: "BCDFGHJKLMNPQRSTVWXYZbcdfghjklmnpqrstvwxyz"
   b: parse s [
       collect [ any keep [ thru c opt v ]
       keep thru end ]
   ]
   p: "" 
   foreach c b [
       e: last c: to-string c
       either find t e [ p: e ][
           parse c [ any [ copy p v | skip ] ]
           if find u e [ append c lowercase p ]
       ]
       prin c
   ]
]

Haskell, 142 130 bytes

""&
import Data.Char
v=(`elem`"aeiouAEIOU")
s&(x:y:z)|v y=x:s&(y:z)
s&(x:y)|v x=x:[toLower x]&y|isAlpha x=x:s++s&y|1>0=x:s&y
_&x=x

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The initial ""& is a partial application of the (&) function defined later on, and is placed so oddly to make TIO count the bytes in ""&, but not count the bytes that, in a full program, would be needed to assign that to any named value.

Less golfed:

import Data.Char (isAlpha, toLower)

vowel :: Char -> Bool
vowel = (`elem`"aeiouAEIOU")

replace :: String -> String
replace = go "" -- start by carrying no extra vowel
  where go _ "" = ""
        -- special case for "anything followed by vowel" so later cases can ignore next character
        go s (x:y:more) | vowel y = x : go s (y:more)
        go s (x:xs) | vowel x = x : go [toLower x] xs -- update the vowel we're carrying
                    | isAlpha x = x : s ++ go s xs -- non-vowel letter not followed by a vowel
                    | otherwise = x : go s xs -- some non-letter junk, just include it and carry on

There really ought to be a way to do this more concisely with a fold instead of recursion, but I couldn't figure it out.

Perl 5, 68 67 59 bytes

perl -pe '$v="[aeiou])";1while s/($v[^a-z]*[b-z]\K(?<!$v(?!$v/\L$1/i'

Here's a great example of the usefulness of \K, and I can't believe I didn't know about this feature before Dom Hastings pointed it out.

I haven't been able to get the right behavior with just using s///g, so an actual loop seems necessary. (It's possible that the right use of a look-behind assertion could work without an explicit while -- but I haven't found it.)

05AB1E, 34 bytes

vyžMylåil©1V}žPylåžM¹N>èå_Y&&i®«}J

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I take that back I can only shave 3 bytes off this monstrosity... I think I could shave the boolean down, but there MUST be 3 cases. 1 for vowels. 1 for consonants. 1 for the case that a digit/symbol exists.

v                                 # For each...
 y                                # Push current element.
  žM                              # Push lower-case vowels (aeiou).
    ylå                           # Lower-case current element is vowel?
       i©1V}                      # If so, put it in register, set Y to 1.
            žP                    # Push lower-case consonants (b...z)
              ylå                 # Is current char a consonant?
                 žM¹N>èå_         # Push vowels again, is input[N+1] NOT a vowel? 
                         Y        # Did we ever set Y as 1?
                          &&      # All 3 previous conditions true?
                            i®«}  # Concat the current vowel to the current char.
                                J # Join the whole stack.
                                  # '}' isn't needed here, b/c it's implied.
                                  # Implicit return.

SWI-Prolog, 593 bytes

a(S,D):-atom_chars(S,D).
g(_,[],_,-1).
g(E,[E|_],R,R).
g(E,[_|T],I,R):-N is I+1,g(E,T,N,R).
c(A,E):-g(E,A,0,R),R > -1.
v(X):-a('AEIOUaeiou',X).
c(X):-a('BCDFGHJKLMNPQRSTVWXYZbcdfghjklmnpqrstvwxyz',X).
d([_],_,R,R).
d([H|T],_,I,R):-v(V),c(V,H),!,d(T,H,I,[H|R]).
d([H,N|T],V,I,R):-c(C),c(C,H),v(W),c(W,N),!,d([N|T],V,I,[H|R]).
d([H,N|T],V,I,R):-c(C),c(C,H),v(W),\+c(W,N),string_lower(V,LV),!,d([N|T],V,I,[LV,H|R]).
d([H|T],V,I,R):-!,d(T,V,I,[H|R]).
r([],Z,Z).
r([H|T],Z,A):-r(T,Z,[H|A]).
r(S,D):-r(S,D,[]).
m(X,R):-a(X,O),r(O,P),r([''|P],Q),d(Q,'',I,[]),r(I,J,[]),atomic_list_concat(J,R).

Used only built-in predicates (without regex or list manipulating library).

Usage:

?- m('A pnm bnn').
'A panama banana'
true .

Japt -P, 28 bytes

ó@\ctX ©\VtYÃËè\v ?P=D:D¬qPv

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Unpacked & How it works

UóXY{\ctX &&\VtY} mD{Dè\v ?P=D:Dq qPv

UóXY{           }  Split the string between any two chars that don't satisfy...
     \ctX &&\VtY     The first char is a consonant and the second is a non-vowel
mD{                And map...
   Dè\v              If this item is a vowel...
       ?P=D            Assign it to P and return as-is
           :Dq qPv     Otherwise, split the item into chars and join with P lowercased
                       (P starts with "", so beginning consonants are not affected)

-P                 Join with ""

The ó function wins over any kind of regexes.

TSQL, 500 bytes

 CREATE TABLE i (i CHAR(999)); INSERT i VALUES ('The rain in Spain stays mainly in the plain')
 DECLARE @w CHAR(999)=(SELECT i FROM i),@r VARCHAR(999)='';WITH d(n,c,i,q)AS(SELECT n,SUBSTRING(@w,n,1),CHARINDEX(SUBSTRING(@w,n,1),'AEIOUaeiou'),CHARINDEX(SUBSTRING(@w,n,1),'BCDFGHJKLMNPQRSTVWXYZbcdfghjklmnpqrstvwxyz')FROM(SELECT DISTINCT number n FROM master..[spt_values]WHERE number BETWEEN 1 AND LEN(@w))D)SELECT @r=@r+f.c+LOWER(COALESCE(CASE WHEN f.q<>0 AND COALESCE(d2.i,0)=0 THEN SUBSTRING(@w,(SELECT MAX(n)FROM d WHERE i<>0 AND n<f.n),1)END,''))FROM d f LEFT JOIN d d2 ON f.n=d2.n-1 SELECT @r

Table i is used for input

Python 3, 125 bytes

lambda s,v='[^aeiouAEIOU':sub(f'(?<={v}\W\d])(?={v}]|$)',lambda m:sub(f'{v}]','',s[:m.end()])[-1:].lower(),s)
from re import*

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Python 3.6 allows us to (ab)use f-strings to reuse our set of vowels (and for four more characters saved, the beginning of an inverted regex character class) cheaply (a f prefix on each string, then {v} as needed, instead of the '+v+' you'd need with concatenation, or the [^aeiouAEIOU you'd insert literally.

The regex that matches no characters, just a position, avoids issues with the non-overlapping matches normal regexes require, and removes the need to backreference any part of the match; all we use the match object for is to get the slice index we use to find the prior vowel.

Partially de-golfed, it would be something like:

import re

def get_last_vowel(string):
    '''
    Returns the lowercase version of the last vowel in a string if
    the string contains any vowels, otherwise, return the empty string
    '''
    try:
        *restvowels, lastvowel = re.sub(r'[^aeiouAEIOU]', '', string)
    except ValueError:
        lastvowel = ''  # No vowels in string
    return lastvowel.lower()

def rememebere_tehe_vowelese(string):
    '''Inserts the lowercased last vowel seen after any consonant not followed by a vowel'''
    return re.sub(r'(?<=[^aeiouAEIOU\W\d])(?=[^aeiouAEIOU]|$)',
                  lambda match: get_last_vowel(string[:match.end()]),
                  string)

Powershell, 104 bytes

based on Neil's regular expression.

[regex]::Replace($args,'(?i)(?<=([aeiou]).*?[^\W\d_aeiou])(?![aeiou])',{"$($args.Groups[1])".ToLower()})

save it as get-rememebere.ps1. Script for testing:

$test = @"
AN EXAMPLE WITH A LOT UPPERCASE (plus some lowercase)
And here comes a **TEST** case with 10% symbols/numbers(#)!
This is an example string.
abcdefghijklmnopqrstuvwxyz
A pnm bnn
Tell me if you need more test cases!
"@

$expected = @"
ANa EXAMaPaLE WITiHi A LOTo UPuPEReCASE (pelusu some lowerecase)
Anada here comese a **TESeTe** case witihi 10% siyimiboloso/numuberese(#)!
Thisi isi ana examapale seterinigi.
abacadefegehijikiliminopoqorosotuvuwuxuyuzu
A panama banana
Telele me ifi you neede more tesete casese!
"@

$result = .\get-rememebere.ps1 $test
$result -eq $expected
$result

sed 4.2.2, 64 bytes

:
s/(([aeiou])[^a-z]*[b-df-hj-np-tv-z])([^aeiou]|$)/\1\l\2\3/I
t

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JavaScript, 88 82 Bytes

Done with a single regular expression:

Original Version (88 Bytes):

s=>s.replace(/(?<=([aeiou]).*?(?![aeiou])[a-z])(?=[^aeiou]|$)/gi,(_,c)=>c.toLowerCase())

Updated Version (82 Bytes) after looking at Neil's regular expression:

s=>s.replace(/(?<=([aeiou]).*?[^\W\d_aeiou])(?![aeiou])/gi,(_,c)=>c.toLowerCase())

var tests = {
  "AN EXAMPLE WITH A LOT UPPERCASE (plus some lowercase)":
    "ANa EXAMaPaLE WITiHi A LOTo UPuPEReCASE (pelusu some lowerecase)",
  "And here comes a **TEST** case with 10% symbols/numbers(#)!":
    "Anada here comese a **TESeTe** case witihi 10% siyimiboloso/numuberese(#)!",
  "This is an example string.":
     "Thisi isi ana examapale seterinigi.",
  "abcdefghijklmnopqrstuvwxyz":
    "abacadefegehijikiliminopoqorosotuvuwuxuyuzu",
  "A pnm bnn":
     "A panama banana",
  "Tell me if you need more test cases!":
     "Telele me ifi you neede more tesete casese!"
};

for ( test in tests )
{
  var result = (s=>s.replace(/(?<=([aeiou]).*?[^\W\d_aeiou])(?![aeiou])/gi,(_,c)=>c.toLowerCase()))(test);
  console.log( result === tests[test], result );
}

Python 2, 134 119 bytes

def f(s,v=''):u=s[:1];return s and u+v*(u.isalpha()-g(u)-g((s+u)[1]))+f(s[1:],[v,u.lower()][g(u)])
g='aeiouAEIOU'.count

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EDIT: 15 bytes thx to Lynn

Standard ML, 225 223 bytes

str o Char.toLower;fun?c=String.isSubstring(it c)"aeiou"fun g(x,l)$d=(fn l=>if Char.isAlpha$andalso not(?d)then if? $then(it$,l)else(x,l^x)else(x,l))(l^str$)fun f$(c::d::r)=f(g$c d)(d::r)|f$[c]= #2(g$c c);f("","")o explode;

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Less golfed:

val lower = str o Char.toLower

fun isVowel c = String.isSubstring (lower c) "aeiou"

(* c is the current char, d is the next char, x is the last vowel and l the accumulator 
   for the resulting string *)
fun g (x,l) c d = 
    if Char.isAlpha c andalso not (isVowel d)
    then if isVowel c 
         then (lower c, l^str c)
         else (x, l^str c^x)
    else (x, l^str c)

fun f t (c::d::r) = f (g t c d) (d::r)
  | f t [c] = #2(g t c #"d")

val h = f ("","") o explode;

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Perl 6,  75 73 71  69 bytes

{({S:i/.*(<[aeiou]>).*<-[\W\d_aeiou]><()><![aeiou]>/$0.lc()/}...*eq*).tail}

Try it

{({S:i{.*(<[aeiou]>).*<-[\W\d_aeiou]><()><![aeiou]>}=$0.lc}...*eq*).tail}

Try it

{({S:i{.*(<[aeiou]>).*<:L-[_aeiou]><()><![aeiou]>}=$0.lc}...*eq*).tail}

Try it

{({S:i{.*(<[aeiou]>).*<:L-[_aeiou]><(<![aeiou]>}=$0.lc}...*eq*).tail}

Try it

Expanded:

{  # bare block lambda with implicit parameter $_

  (
    # generate a sequence

    {  # code block used to generate the values

      S               # substitute (not in-place)
      :i              # :ignorecase
      {

          .*              # start at end of string

          ( <[aeiou]> )   # store the previous vowel in $0

          .*

          <:L - [_aeiou]> # letter other than a vowel

          <(              # ignore everything before this

                          # this is where `$0.lc` gets inserted

          # )>            # ignore everything after this

          <![aeiou]>      # not a vowel (zero width lookahead)

      } = $0.lc       # replace with lowercase of the earlier vowel
    }

    ...    # keep generating until:

    * eq * # there are two equal strings (no changes)

  ).tail   # get the last value
}

JavaScript (ES6), 108 105 bytes

(Saved 3 bytes thanks to @Shaggy.)

f=s=>(t=(s+=' ').replace(/[aeiou]|[a-z][^aeiou]/ig,r=>r[1]?r[0]+v.toLowerCase()+r[1]:v=r,v=''))!=s?f(t):s

Searches for vowels or for consonants with no following vowel:

/[aeiou]|[a-z][^aeiou]/ig

(We don't need to search for consonants explicitly, because vowels are excluded based on the /[aeiou]|....)

Vowels are stored in v, and consonants with no following vowel have v inserted:

r[1]?r[0]+v.toLowerCase()+r[1]:v=r

(If r[1] exists, we've matched on a consonant plus non-vowel.)

If nothing has been changed, we return the input. Otherwise, we recurse on the replaced string.

f=s=>(t=(s+=' ').replace(/[aeiou]|[a-z][^aeiou]/ig,r=>r[1]?r[0]+v.toLowerCase()+r[1]:v=r,v=''))!=s?f(t):s

console.log(f('A pnm bnn'));
console.log(f('abc'));
console.log(f('AN EXAMPLE WITH A LOT UPPERCASE (plus some lowercase)')); 
console.log(f('And here comes a **TEST** case with 10% symbols/numbers(#)!'));
console.log(f('This is an example string.'));
console.log(f('abcdefghijklmnopqrstuvwxyz'));
console.log(f('Tell me if you need more test cases!'));

JavaScript ES6, 115 bytes

Saves 8 bytes thanks to @ETHProductions

s=>[x="",...s].map((i,j)=>(r=/[aeiou]/i).test(i)?x=i:/[a-z]/i.test(i)&&!r.test(s[j]||1)?i+x.toLowerCase():i).join``

I have managed to inflate this more in the process of golfing it O_o but it also fixes a bug

s=>[x="",...s].map(             // Create a new array with x storing last vowel
                                // This also offsets indexes by one so rel to original str refers to next char
   (i,j)=>                      // Going through each char...
      (r=/[aeiou]/i).test(i)?   // If it's vowel, store it in x
          x=i:
      /[a-z]/i.test(i)          // If a letter (thats not a vowel excluded by above)
         &&!r.test(s[j]||1)?    // Test if next char is *not* vowel
         i+x.toLowerCase():i    // If it isn't, then add the most recent vowel after
    ).join``                    // Combine back to string

JavaScript (Node.js), 146 143 132 127 125 bytes

(a,v="")=>[...a].map((b,i)=>(w=/[aeiou]/i).test(b)&&(v=b)?b:w.test(a[i+1]||b)||!b.match(/[a-z]/i)?b:b+v.toLowerCase()).join``

Try it online!