I was wondering what your reasoning was for animating the vertices rather than just moving the object or just creating shape keys. Although you can animate various parameters in Blender in edit mode manually or using the Anim All plugin.
I have tried to get this sort of animation to work in 4.1.1 but it will not. The Puzzles will recognize the group but will not pickup the individual vertices associated with that group from what I can tell.
I am also curious to hear back from Soft8Soft. With your test scene that has 275 vertices that are animating I am wondering what the performance efficiency would be vs just animating the object. Seems like moving a single origin point for a single object would be more efficient but I could be wrong…… waiting and watching.