The Shape Painter tool does something close to what you want, I think.

With the shape selected, go to that Shape painter and choose "Warp" from the dropdown at the right hand end. Then drag the point in smooth strokes across the shape in the direction you want it to move.

So long as you cross both edges of the shape at roughly the same speed they will move by the same amount, thus keeping the profile.

(Sounds more technical than it really is.)