Javascript, 854 bytes

function f(w){h=[];z=[];for(y=0;y<9;y++){l=w.split('\n')[y];r={s:parseInt(l[0]),d:1,e:[]};for(x=0;x<9;x++){c=l[1+x];if(c=='v'||c=='l')r.d=-1;r.e.push(c);}z.push(r);}h.push(z);h.push(z);for(g=2;g<40;g++){z=JSON.parse(JSON.stringify(z));for(y=0;y<9;y++){r=z[y];if(r.s>0&&(g%2==0||r.s==2)){i=0;t=0;if(r.d==-1){t=r.e[0];for(i=0;i<8;i++)r.e[i]=r.e[i+1];r.e[i]=t;}else{t=r.e[8];for(i=8;i>0;i--)r.e[i]=r.e[i-1];r.e[i]=t;}}}h.push(z);}j=15;k="";m(0,4,8,"","");return k==""?"N":k;function m(s,a,b,u,v){if(s<j){q={'U':[0,-1],'D':[0,1],'L':[-1,0],'R':[1,0]}[u];if(q)a+=q[0],b+=q[1];v+=u;if(!b){j=s;k=v;return;}if(s>0){for(i=0;i<3;i++){z=h[s*2+i-2];r=z[b];if(i&&(i==1||r.s==2)&&r.e[a].toUpperCase()=='L')a+=r.d;if(a<0||a>8||b>8)return;z=h[s*2+i-1];e=z[b].e;if(e[a].toUpperCase()=='V'||(b<4&&e[a].toUpperCase()!='L'))return;}}t="UWRLD";for(x in t)m(s+1,a,b,t[x],v);}}}

Ungolfed

function solve(input) {
    var grids = [];
    var maxgrid = 40;
    // load input
    var grid = [];
    var lines = input.split('\n');
    for (var y = 0; y < 9; y++) {
        var line = lines[y];
        var row = {
            speed: parseInt(line[0]),
            direction: 1,
            cells: []
        }
        for (var x = 0; x < 9; x++) {
            var c = line[1 + x];
            if (c == 'v' || c == 'l')
                row.direction = -1;
            row.cells.push(c);
        }
        grid.push(row);
    }
    grids.push(grid);
    grids.push(grid);
    // animate grids
    for (var g = 2; g < maxgrid; g++) {
        grid = JSON.parse(JSON.stringify(grid));
        for (var y = 0; y < 9; y++) {
            var row = grid[y];
            if (row.speed > 0 && (g % 2 == 0 || row.speed == 2)) {
                if (row.direction == -1) {
                    var i, temp = row.cells[0];
                    for (i = 0; i < 8; i++)
                        row.cells[i] = row.cells[i + 1];
                    row.cells[i] = temp;
                }
                else {
                    var i, temp = row.cells[8];
                    for (i = 8; i > 0; i--)
                        row.cells[i] = row.cells[i - 1];
                    row.cells[i] = temp;
                }
            }
        }
        grids.push(grid);
    }
    var best = 15;
    var best_moves = "";
    var forceExit = false;

    move(0, 4, 8, "", "");
    return best_moves == "" ? "N" : best_moves;

    function move(step, fx, fy, dir, moves) {
        if (step >= best)
            return "die";
        switch (dir) {
            case 'U':
                fy--;
                break;
            case 'D':
                fy++;
                break;
            case 'L':
                fx--;
                break;
            case 'R':
                fx++;
                break;
        }
        if (dir != '')
            moves += dir;
        if (fy == 0) {
            best = step;
            best_moves = moves;
            return "win";
        }
        if (step > 0) {
            for (var i = 0; i < 3; i++) {
                var grid = grids[step * 2 + i - 2];
                if (i > 0 && (i == 1 || row.speed == 2)) {
                    var row = grid[fy];
                    if (row.cells[fx].toUpperCase() == 'L')
                        fx += row.direction;
                }
                if (fx < 0 || fx > 8 || fy > 8)
                    return "die";
                var grid = grids[step * 2 + i - 1];
                var cells = grid[fy].cells;
                if (cells[fx].toUpperCase() == 'V')
                    return "die";
                if (fy < 4 && cells[fx].toUpperCase() != 'L')
                    return "die";
            }
        }
        move(step+1, fx, fy, 'U', moves);
        move(step+1, fx, fy, 'W', moves);
        move(step+1, fx, fy, 'R', moves)
        move(step+1, fx, fy, 'L', moves);
        move(step+1, fx, fy, 'D', moves)
    }
}

I'm only testing for Up, Wait and Right in the example to speed things up:

JSFiddle