I think it could be the the case that Luxu's memories brought them back... Though it's a bit confusing. You can't just bring ANYBODY back by just remembering them, right? We know that with the whole Ark concept, that peoples hearts can be sent to a certain time and location, though their bodies fade away... And if someone is around at this destination to help people remember you, somehow, your body can then be restored.I'm seen some very interesting theories as of late regarding the black box with the latest union cross updates, and all so good that I'm not sure which one to follow.
One theory - that the black box itself is the ark. I believe this theory belongs to PodCast Now.
One theory - the black box itself is the intermediary object that is mentioned by darkness. Maybe it holds the souls or hearts of the foretellers? The box itself was buried in the keyblade graveyard. Reason foretellers are back is because Luxu is the one to remember them. So maybe they used the ark? I think this was a theory of GamersJoint on youtube. This is one I really want to believe the most. It would explain why the box looks so dirty in the scene.
Were the hearts of the Foretellers just lying around in the Keyblade Graveyard, waiting for someone to remember them? Could be. Seems weird though. Maybe Luxu collected their hearts in the Black Box after the original keyblade war, and the box is essentially just a vessel to store hearts which can then be used restore them when necessary?
Ava must still either exist somehow, or had her heart released somewhere else, and that's why she doesn't return when Luxu remembers her.