First I'd like to say that I'm a subscriber to the reincarnation theory. But I feel there's is a hole in this theory that goes largely ignored. Kairi. Here I'm going to detail the problem and see of we can't find a way the theory works.
As stated in 3D, when the χ-blade shattered, the 7 fragments of light eventually became the hearts of the Princesses of Heart. Kairi, is one of these Princesses. This means that, before the Keyblade War, her heart was part of the χ-blade. Therefore Kairi could not have existed as an individual prior to the χ-blade shattering. If she did not exist as a person before the Keyblade war, she can not be the reincarnated Ava.
So, does anyone have a way Nomura could work around this? It's possible I suppose for Kairi's Grandmother to have been Ava in a past life, but not Kairi herself.