Warp points binding in After Effects
If you haven't found it yet, the Puppet tool likely what you're looking for.  You will create specific anchor points and you can keyframe their motion. 

There's several tutorials out there for the right way to use the tool, but the best advice I can offer is to just play with it till you get the desired effect. 

Personally I put more points where I DONT want it to move and use fewer points where I do.    To practice just create a solid and drop points on it, then grab a point and start moving it around.   It will become clear.