If I’m allowed to make an analogy:
Trying to assemble this model may be like putting together pieces of a broken egg shell that should have some kind of movement.
If they are not supported from behind in some way, this can be a very difficult task.
Two solutions come to my after reading the comments by Zathros and Chuck: 1) the use of different kinds of paper (different thickness) to make some inner “flexible” joints (I was thinking on “V” struts); or 2) the use of some sort of nucleus to which the continents are “fixed” from the underside, allowing their borders to have some freedom of movement.
It would make it a lot easier to activate the movement of most of the plates (or at least their borders), although the additional work is also bigger.
As soon as I have some time I’ll continue with it