How to stretch an object in After Effects?

That could be quite tricky, since AE doesn't have a simple path deformer or "cvonnect to spline" function. Either you slice it up in segments and then rig it up with the respective functions to follow a mask path based on expressions or you use distortion effects like Mesh Warp, Bezier Warp or the Puppet tool and animate their handles. The trick with everything is obviously that correct alignment in the initial pose is of the essence and that you may have to split up your work into segments regardless in order to avoid things getting crinkly.

Mylenium