(covers information from several alternate timelines)
Encounters with Voyager
Braxton was first encountered by the USS Voyager in 2373, when the Aeon arrived in the Delta Quadrant through a temporal rift. Braxton claimed that Voyager had been responsible for the destruction of the Sol system in the 29th century, and had traveled back to prevent that occurrence by destroying Voyager. The Voyager crew was able to counter Braxton's attack, however, destabilizing his temporal rift, into which both ships were pulled. Both ships were subsequently sent back to Earth, the Aeon to 1967, and Voyager to 1996.
Braxton managed to execute an emergency beam-out before crashing in the High Sierras, but was unable to retrieve his ship before it was discovered and stolen by a camper named Henry Starling. Over the next thirty years, Starling would utilize his primitive understanding of the timeship's technology to introduce new computer innovations to the public, constructing a corporate empire and initiating the computer revolution of the 20th century.
When Voyager arrived in Earth orbit in 1996, they discovered Braxton living homeless on the streets of Los Angeles. Poor treatment at the hands of the 20th century authorities had left him paranoid and aggressive, blaming Voyager for his predicament. Braxton informed Captain Janeway of his deduction that, if Starling were to use the timeship to travel to the 29th century without properly recalibrating the ship's temporal matrix, then that could have been the cause of the temporal explosion which destroyed Earth's solar system. The crew of Voyager was able to prevent this future by destroying the Aeon with a manually-launched photon torpedo just before Starling entered the temporal rift, resulting in the creation of a new timeline in which the explosion never occurred. Shortly afterwards, Voyager encountered another temporal rift, created by Captain Braxton's counterpart from the new timeline. Having detected Voyager's anomalous presence in the 20th century, Braxton had traveled back in time to escort Voyager back to their proper place and time. Despite Janeway's plea to return them to Earth, as Voyager had been stranded in the Delta Quadrant for two years, Braxton insisted on upholding the Temporal Prime Directive. (VOY: "Future's End", "Future's End, Part II")
After his thirty-year exile on Earth, Braxton was forced to undergo extensive rehabilitation before he was allowed to return to duty as captain of the Relativity. In that capacity, he recruited Seven of Nine from the year 2375 to a mission to save Voyager from destruction. Bringing Seven aboard the Relativity, he explained that a saboteur had placed a temporal disruptor aboard Voyager, and that they needed her to identify the individual responsible and prevent the disruptor from destroying the ship. During the course of her investigation, Seven discovered that the saboteur was, in fact, Captain Braxton's counterpart from a future point in time. Braxton's counterpart explained that, in the upcoming few years of Braxton's life, he would be sent into rehabilitation again due to suffering from temporal psychosis, and subsequently forced to retire. Claiming Captain Janeway responsible for these events, due to the repairs he had to effect in response to the various time travel events in which Voyager was involved, including the temporal inversion in the Takara sector, he became determined to wipe Voyager from the timeline. Braxton's future counterpart attempted to convince Braxton to let him proceed and, in light of these revelations, Braxton's first officer, Lieutenant Ducane, placed Braxton under arrest for crimes he was going to commit, and took control of the mission. Ducane was eventually successful in apprehending Braxton's counterpart, but was still left with the task of repairing the damage done to the timeline due to Braxton's actions. As Seven could not make another time jump without suffering from temporal psychosis herself, Ducane recruited Captain Janeway to complete the mission. In exchange for leniency, Braxton's counterpart agreed to inform Ducane of the precise time and place where he planted the disruptor. Janeway was then sent to the year 2372, where she was able to apprehend Braxton before he was able to activate the disruptor. This version of Braxton was subsequently transported back to the 29th century, resulting in the existence of no less than three versions of him being held in custody, although Ducane assured Janeway that all three of them would be reintegrated in time for him to stand trial. (VOY: "Relativity")
It is unclear exactly how the later Braxton remembers being trapped on Earth. However, given the continued existence of The Doctor's mobile emitter and the fact that Voyager continues to orbit Earth in 1996 rather than revert to the Delta Quadrant, it seems likely that the Braxton who was trapped for thirty years also continued to exist. One can speculate that he was picked up and underwent temporal integration with the second Braxton before returning to duty.