Can you move a broken finger?

Depends on your tolerance for pain, where the break is, and what you mean by moving it.

Trying to get a limb to bend between joints is a classic way of checking for fractures, so yeah a broken finger can be moved - but it'll be painful.

If the question is about - say making a fist with a broken finger - well, you're unlikely to be able to curl it up real tight, b/c of pain. But you may well be able to wiggle it about a bit.