If time travellers are still around, wouldnt they be capable of moving backwards in time before the fall?
Iam certain if you and me were time travellers we would have made certain of being capable of transporting ourselves forwards or backwards in time, as a "Plan B" ..
More than likely, it would be impractical to transport large chunks of the population to another time period. Part of what causes a civilization to crumble is the mindset it's population's attitudes towards things, for instance, feminism and transgenderism is a recurring theme through civilizations in their later stages. In essence the civilizations are so concerned with some irrelevant detail and so concerned about controlling other peoples thoughts and ideas, that they leave their watchtowers abandoned, meanwhile the barbarians surround them ready to invade.
If you were going to transport people forward you'd be wanting to screen them to make sure they didn't infect the new civilization with pre-fall philosophies that cause the new civilization to falter. You'd also want this as it adds new genetic data into the gene pool, especially useful should you have a small gene pool to work with. (I had a vision that this was an early problem with the civilization that is personified by the man with the scepter)
Technologically speaking though, it'd be fairly difficult to rebuild the technology needed for a temporal jump to achieve the technology needed to rebuild advanced technology. Ideally you'd want to use pre-fall tech to aquire the tech as it'd be more advanced than a small population could come up with, and thus grant a larger range. Though realistically, there would be a limit to how far this would work for until they'd be faced with a sliding range problem. An initial machine might do say, 10 years from 2036. If the west falls in 2022, then you'd be 4 years too late. In another 2 years they get to say, a 20 year range. They'd be within 3 years of the fall. Say progress slows down, in 2060 they can only get a 35 year range, they'd be 3 years late.
Not to mention, crude displacement technology might possibly leave damaging effects that we can't detect yet...