So far it never fails.
There may be some odd comments in the process, but it always worked for me. I've also successfully moved other villager houses (not the one being replaced) and build ramps/destroyed bridges during this process.
I too was Time Traveling, but always 1 day at a time - since you get the perfect rhythm of having the old villager instantly in boxes headed out [Day 0, visit 3 on first amiibo], an empty plot reserved for the first amiibo [Day 1, visit 1 of second amiibo], first amiibo present in new house with boxes (moving in) [Day 2, visit 2 of second amiibo], and then greeting your first amiibo when they start walking around town and instantly boxing another old villager [Day 3, visit 3 of second amiibo].
I've scanned amiibo at all times of the day - sometimes at just after 5AM all the way as late as 2AM with no notable differences. Some tasks required traveling to Nook Islands for additional materials, but that didn't affect things at all (related and possibly notable: I never failed to bring a requested item. Failing theoretically could factor in to issues with move-ins, but I cant confirm or deny). Also possibly of note is I have not updated to v1.1.1 yet (I'm still on v1.1.0 for the time being).
It's probably unimportant, but all my amiibo cards were unregistered (no owner info/game data/etc.) as if they just came out of the package. If there's any other info I can supply to help successful moves, let me know.