Here's what happened, as far as I understand it:
Luxu meets Brain and removes his heart from his body, placing it in the ark to send it forward in time. Luxu then possesses Brain's heartless body and uses it as the first of his many vessels. Brain's heart arrives at an undisclosed point in time, most likely sometime in between UX and Dark Road when Scala ad Caelum looked very different, and once there, Sigurd resurrects his body via his hat and also his own knowledge of Brain, probably handed down through the ages by many Keyblade Masters after Ephemer.
That's about it. If there's anything still unsolved, let me know.
The main thing debunking this is secret report #12, which clearly says that Luxu gave away his keyblade before doing any body-hopping, but in the face reveal scene he still has it. Another is secret report #11, where the talk about the virus and "the same trick" has no explanation in this theory, but starts making sense if you assume that Luxu himself is the virus.
This is what I've always assumed as well, however, a really solid counter point has been - why would Luxu need/want take over Brain's body? Luxu isn't shown to be old or anything or in need of a new body yet...
This is also true.
Also, why did Luxu restore Strelitzia and send her to the future? Luxu as MoM's disciple has no reason to do that, as it is against his instructions. But Brain of course would do it, for his friend Lauriam.
You're brilliant in every sense of the word.
It'd be a waste if your life were to end here.
Just for clarification, this wording is from the unofficial translation. The official one is quite lame compared to it. Isn't "brilliant in every sense of the word"
as a catchphrase brilliant in every sense of the word? 😆
A hypothetical question, would people here be interested in reading a piece of fanfiction where Brain himself explains what is going on behind the scenes in UX?