The basic rules of how time travel works in KH universe:
1) To travel through time you must cast aside your body, because only your heart is able to make the journey.
2) One can only travel to points in time where one has an 'anchor' of sorts--a version of oneself that exists there already.
3) When one returns to their original time, they will forget the experience and all memories of it, but their heart will have a sort of inkling that it happened, without the person really knowing why. This can guide them to take certain actions.
4) It has been established that even if you don't have a body, so long as someone remembers you and you are meant to exist in that time, your body is able to be reconstructed around your heart.
*Timeless river seems to operate outside of these rules, and it just sort of...does its own thing.
The Ark in UX works on a similar level. First, there were technically two separate arks, one in the datascape that transported everyone from inside the datascape back to the real world, and the other ark (the "real" Ark) which sends them to some random point in time in the future. As far as "Darkness" explained, while riding in the ark your flesh is stripped away leaving just a heart. Then, when you reach your destination some "intermediary" meets with you to reconstruct your body. And, if people remember/know of you in that time period, your body will be restored. As far as I know, there isn't any more information as to who or what that "intermediary" is, but perhaps it has something to do with the people who contacted Brain upon his exit of the pod.
Either way, events that have happened are not meant to be changed by time travel. If something happens, even if you time travel it will still happen because the time traveling allowed for those events to happen. However, the Master of Masters mentioned about branching/alternate timelines. The reason why the MoM trapped the Union Cross people inside the digital space was to prevent them from using time travel to create an alternate time-line using knowledge from the book of prophesies.
This essentially prevented alternative timelines from being an issue UNTIL Sora blew that up during the Keyblade War. When Sora used the power of waking to save everyone that DID create an alterate timeline. The first real split. Because the original predicted timeline was that they all were defeated, which DID happen. Then Sora created a split in the timeline to create a new one where they were not defeated.
During Re:mind Sora used the power of waking again to attempt to bring Kairi back. This didn't create an alternate timeline. Sora went back in time (as a heart) and was essentially taking refuge inside of his friends' bodies as they went through the events of the keyblade war again. This allowed him (and us as the audience) to witness other perspectives of the same fights, but those events always played out that way the first time, we just weren't able to see it because Sora was either not there or unconscious. In other words, he wasn't actually changing anything in the past here.
As far as the Sora and Kairi vs Xehanort fight goes-- the original final boss fight was against Xehanort's heart. We sort of went into that portal thing and Sora was kind of beating him up from the inside, basically. But Xehanort's body would still have been outside where all the other Guardians were. It is my understanding that Sora and Kairi were beating up Xehanort's body outside while original kh3 Sora, Donald and Goofy were beating up his heart on the inside.
Eventually once Sora catches back up to the time that he first left to go find Kairi then the points in time would merge back together. As far as I know there weren't any alternate timelines created here, because as far as we saw with the vanilla kh3 ending, Kairi was always saved. However, the act of saving her would have at the very least left an imprint on their hearts of the experience (it's not clear how their memory of the events were impacted, but we know at least Kairi remembers the Final World).