[Jeppe Rasmussen]

Several ways to do this:

If you want the layer to actively move around.
You can create a null object, parent your layer to this null and then move/keyframe the null around to get the movement you want.

If you want the layer to stay in place but at a different position
Select ALL of the keyframes of the property that you want to change (Click on the property name).
Make sure the timeline marker is on top of one of the selected keyframes.
Move the layer.

You could also precomp the layer and then move the precomp, but first you would have to solve scaling and clipping issues caused by the stabilizing.

I’d probably go with the null, myself.

Happy holidays

[Jeppe Rasmussen]

First of all: This might well be super fancy 3D, but keeping in mind that the effect only lasts for a couple of seconds and that the perspective change and skin distortion is minimal, it could easily be done in after effects.
You’d have to track a premade grid and colour to the actors face, and then mask out the hand.

[Jeppe Rasmussen]

Also, when you get the layers set up, remember that if they intersect or touch in 3d space you’ll want to set their blend mode to “alpha add”. That’ll get rid of those annoying artifacts along the edge.