
Fusion is capable of doing it. If you stick to multiplane without organic deformations, Fusion will work for you pretty well. But if you're animating and deforming, it might not be as easy as doing it in AE with the puppet tool.
I'd say your best bet in Fusion would be using the 3D system for the depth, the grid warp for the deformations. Actually, you could add some real depth to some elements with some well painted gradient and 3D Displace tool.