Randomly change position from a given point in composition

Wiggle could also be an easy solution. Set the frequency to 0, to make sure it is not wiggling at all:

This example will move the layers up to 200px in a random direction;

the move starts at 1s and ends at 3s:

var amplitude = linear(time,1,3,0,200)
wiggle(0,amplitude)

As imple as using a linear interpolator:

mPos=thisLayer.transform.position;

mTime=2;

seedRandom(index,true);

randX=random(0,500);

randY=random(0,200);

X=linear(time,0,mTime,mPos[0],mPos[0]+randX);

Y=linear(time,0,mTime,mPos[1],mPos[1]+randY);

[X,Y]

This can of course endlessly be modified and expanded upon, but that would require a deeper explanation as to how you actually want it to work rather than such a borad and generic "Can I do this...?" question, in particular when it comes to the specifics of the timing or where you want you layers to end up in the comp.

Mylenium