JavaScript (ES6), 100 90 88 bytes

f=s=>s==(l=s.search` `,s=s.replace(eval(`/(\\S)([^]{${l-1},${l+1}}) /`),` $2$1`))?s:f(s) s=` (__) (oo) /------\\/ / | || * /\\---/\\ ~~ ~~ ..."Have you mooed today?"...` console.log(s) console.log(f(s))

Requires the the string to have at least two lines and all lines padded to equal length. Output for the example image:

 ( -- /|---/|-(o__ */~~\---~\|\/o)) ..."Have you/mooed~today?"... 

Note that as it tries to move elements to the right if possible, the * did not fall between the Have and the you.

Edit: Saved 10% thanks to @ETHproductions. Saved another 2 bytes thanks to @DanielIndie.

Retina 0.8.2, 50 bytes

+`(?<=(.)*)(\S)(.*¶(?<-1>)?(?>(?<-1>.)*).?) $3$2 

Try it online! A slightly different approach to my JavaScript answer, this uses a balancing group to match a space below the non-space character; the (?<-1>)? allows the space to be one column to the left while the .? allows the space to be one column to the right.

Retina, 40 bytes

~0L$`.(.*)¶ +s`(\S)(.{$.1,$.&}) ¶ $$2$$1 

Try it online! Port of my JavaScript answer. The 0L$ atomic stage takes the input and substitutes the two lengths into the second line resulting in the command that actually performs the replacement, which is then evaluated on the original input by the ~ compound stage.

Python 2, 298 bytes

a=input() L=len(a);s=' ' a=[list(s*L+l.ljust(L+max(map(len,a))))for l in a] t=1 while t: t=0 for y in range(L-1): for x in range(len(a[y])): c=a[y][x];C=a[y+1][x-1:x+2] if s!=c and s in C:t=1;a[y][x]=s;a[y+1][[[x+1,x][C[1]==s],x-1][C[0]==s]]=c for l in map(''.join,a):print l[L:].rstrip() 

Takes input as a list of strings (one per line)

Example: Input:

[' (__)', ' (oo)', ' /------\/', ' / | ||', ' * /\---/\ ', ' ~~ ~~', '..."Have you mooed today?"...'] 

Output:

 ( -----/|-(o__ //|~~\---~\|\/o)) ..."Have*you/mooed~today?"... 

C, 252 bytes

e=1,l,c,i,j,p,r,w,a[999];f(){while((i=getchar())>0)a[w++]=i,i<16?l++:0,l?0:c++;while(e)for(i=e=0;i<c;i++)for(j=l;j>=0;j--)e=(r=a[p=j*(c+1)+i]-32?a[r=p+c+1]-32?a[r=p+c]-32?a[r=p+c+2]-32?0:r:r:r:0)?l=a[p],a[p]=a[r],a[r]=l:e;for(i=0;i<w;)putchar(a[i++]);} 

Ungolfed test code:

#include <stdio.h> e=1,l,c,i,j,p,r,w,a[999]; f() { // counting lines and columns while ((i = getchar())>0)a[w++] = i, i<16 ? l++ : 0, l ? 0 : c++; // main shaking loop while (e) // repeat while collapsing for (i = e = 0; i < c; i++) // columns loop for (j = l; j >= 0; j--) // lines loop e = ( // remember that collapsing was r = // find place to collapse a[p = j*(c + 1) + i] - 32 ? // if not space a[r = p + c + 1] - 32 ? // if char under the current is not a space a[r = p + c] - 32 ? // see one position left a[r = p + c + 2] - 32 ? 0 // then one position right : r : r : r : 0 ) ? // and if place was found l=a[p],a[p]=a[r],a[r]=l // replace values in positions p and r : e; //print resulting picture for(i=0;i<w;)putchar(a[i++]); } int main(void) { int cnt; FILE * testf = fopen("caw.txt","w"); char testd[][31] = { " (__) \n", " (oo) \n", " /------\\/ \n", " / | || \n", " * /\\---/\\ \n", " ~~ ~~ \n", "...\"Have you mooed today ? \"...", "" }; // prepare data for test printf("Initial data:\n"); for(cnt = 0; cnt < 7; cnt++) { printf("%s", testd[cnt]); fprintf(testf, testd[cnt]); } fclose(testf); // redirect standard input freopen("caw.txt", "r", stdin); printf("\n\nResult:\n"); // start test f(); } 

Result of test:

JavaScript, 286 bytes

b=>eval('f=b=>b==null||" "==b;b=b.split`\n`.map(b=>[...b]);a:for(;;){for(c=0;c<b.length-1;c++)for(g=b[c],d=0;d<g.length;d++){h=g[d];if(!f(h)){e=0;f(b[c+1][d])?e=2:f(b[c+1][d-1])?e=1:f(b[c+1][d+1])&&(e=3);if(e){b[c+1][d+e-2]=h;b[c][d]=" ";continue a}}}break}b.map(b=>b.join``).join`\n`') 

Examples

// Here I assume that you've assigned the above function to `fall` console.log(fall(` (__) (oo) /------\/ / | || * /\---/\\ ~~ ~~ ..."Have you mooed today?"...`)) 

Output:

 - /--(-\--(__ /|~~---/~||/oo)) ..."Have*you/mooed~today?"... 

Another example:

console.log(fall(` (\__/) .~ ~. )) /O O ./ .' {O__, \ { / . . ) \\ |-| '-' \ } .( _( )_.' '---.~_ _ _&`)) 

Output:

 _ , / OO/__'_.. . {.(|-|.(O'))/.~{ /('---.~___-_&)_.'}\~.')) 

Ungolfed function

function fall(input) { let move = true let lines = input.split("\n").map(line => line.split("")) let isSpace = c => c == null || c == " " loop: for (;;) { for (let y = 0; y < lines.length - 1; y++) { let line = lines[y] for (let x = 0; x < line.length; x++) { let ch = line[x] if (!isSpace(ch)) { let dx = 0 if (isSpace(lines[y+1][x])) { dx = 2 } else if (isSpace(lines[y+1][x-1])) { dx = 1 } else if (isSpace(lines[y+1][x+1])) { dx = 3 } if (dx) { lines[y + 1][x + dx - 2] = ch lines[y][x] = " " continue loop } } } } break } return lines.map(line => line.join("")).join("\n") }