Why in boxes, though?
By the time a villager is in boxes, it's been 4 or 5 days since they first decided to move out. It's like the final stage of the moving out process if you will. It wouldn't make much sense if by resetting you could change who was on that final stage, since who ends up being in boxes or not is something decided based on things that took place days before (namely, them pinging you, or just assuming they can move). If anything, resetting could work for the day when someone is supposed to ping you, by making you able to reset who's going to ping you on that day, but unfortunately I don't think that's the case either.
- - - Post Merge - - -
What if they never pinged you and it was a matter of, say, time traveling a year into the future so someone would move out?
As long as you don't ever load up your mayor in the future , I think there is a method of cycling that uses this. But I can't remember which one it is!
The method of moving into the future, like Jesusfreakette said, doesn't involve loading up your mayor after you TT. I can give you more details since I've used it and it's my preferred method (since it's pretty safe in my experience, and has worked very well for me) but it's basically along the lines of this: you TT one year into the future, start a new character, check who's gone, go back one year, then use your villager to interact with that villager (talk to them several times if they don't ping you, and if they don't mention moving, save and quit and go talk to them again). If it's someone you want to keep, you'll ask them to stay, then TT one year into the future with your mayor, save and quit, TT another year ahead using a new character to check who's gone, go back by one year, use your mayor to get that character to ping you, etc.
There's more details to it, and I can explain it better, but those are the basics. You don't TT ahead with your mayor, you TT ahead with a new character (using the 3DS clock to adjust the time) to check who's gone and quit without saving.
In this method I've vaguely outlined, they DO ping you, as I explained above. If they don't ping you on your first try and don't bring it up in conversation, just save and quit and try again (without TTing) as many times as you need until they do.
