How do I link an animation to the duration of a layer?

You have to set up an animation that uses the layer in-point and out-point as drivers for an interpolation method. Something like this:

y = thisComp.height/2;
Xstrt = - sourceRectAtTime().width/2;
Xmid = thisComp.width/2;
Xend = thisComp.width - Xstrt;
t = time - inPoint;
dur = .5;
if (t < dur){
	x = easeIn(t, 0, dur, Xstrt, Xmid);
}
else{
	x = easeOut(t, outPoint - dur, outPoint, Xmid, Xend);
}
[x, y]

That expression will give you a half-second slide in and slide out based on the layer in and put points. It will work with shape, text, or footage layers and comps as long as you don't scale the layers. If you need to scale the layers, you will need to throw in a scale multiplier for the start and end positions. The anchor point must be at the default middle of the layer position, and a text layer needs to be center justified unless you throw in a sourceRectAtTime.left value.