Spine. What is the name of the process?
This appears to be simply skeletal animation, which is a standard technique that is available in all modern animation packages. Whether applied to 3D meshes or (as here) 2D ones, the principle is the same.

The pseudo-3D effect on the sprites is created by using a mesh where vertices are placed along the contours of the sprite texture, then animated to squash and stretch parts the image in such a way that it creates the illusion of 3D motion, even though the sprite images are authored in 2D. There's a description of the technique in the Spine docs.