Wow... Okay, this is quite a peculiar phenomenon - particularly where one person tested 100 days of time travel and never had the villagers move out, but other people confirming they've asked to move out in only a week...
...Well, with that observation, this tells me one big thing: There may be some kind of unknown requirement to making them move!
I can't say exactly what it is (it's unknown), but after looking at the time traveling technique I saw in these forums, it looks like ignoring them and passing time makes them stay around. Simply put, doing nothing with them doesn't satisfy the requirement for them to move.
For those who had pings for them to move out... How exactly do you interact with the amiibo villagers compared to non-amiibo ones? This could help solve the issue at hand.