Ok, here's the method I use... I changed up some of the stuff to take into account the fact that you're cycling.
The method normally starts when a villager pings you - then you can deny a request to move and start by going to your 3DS settings and changing the date to the Monday of a week without any events in it. Load as your mayor on that date, then save the game and quit. Return to your 3DS settings and set the date one year forward. Go to your game but this time, start as a new character. The answers you give Rover don't matter since you won't be keeping this character anyway. Just check the town map to see who's missing and quit without saving. Set the 3DS clock back one year again and this time, load as your mayor. Go talk to the villager who would be missing in the future. If they don't immediately ping them, talk to them a few times, save and quit, load the game, and talk to them again. Do that last step until they ask you to move.
If it's a villager you want to keep, say no. Save and quit. Set the 3DS calendar forward by one year, to date you checked with a new character, but this time load the game as your mayor and just save and quit. Set the calendar forward by another year and check who's missing using a new character, then back one year again (to the date you last saved as the mayor) to talk to that villager. Keep doing this until it's a villager you want to move out.
Once it's a villager you want to move out, tell them go. You then two options. You can stay on that year and move them out on their move out date, or you can return to the present date. Just make a note of the date your villager gives you, because if you go back in time, it counts as a day forward and you'll have to do the maths. For example, if your villager mentioned moving in 5 days, they'll now move in 4 days (in case you want to adopt them out).
Then just keep repeating that process. It's very safe as long as you pay attention to not start as your mayor when you're starting with a new character - when you start as a new character, the state of your town doesn't get saved, so if a villager is missing you can fix that. Whereas if you start with your mayor on those steps, it gets locked in. Also, on the steps where you're supposed to use your mayor, nobody will have moved out because on the last time you played, you denied a request to move. Villagers don't move when on your last played date you denied a request.
Now, because you're aiming to move out more than one villager, when you start or when you move out a villager and want to start the process again, it will be slightly different. Just start up the game normally and save and quit. Set the date some months or years into the future and use a new character to check who's missing from the map. If nobody is, just set the date to a Monday of a week with no events in it (game weeks start on Mondays) and load as your mayor (if you want to be extra safe, you can use a new character to check first). If someone is, go back and use your mayor to talk to that character in a way similar to what I described above. Deny that villager's request to move and then change the date to a Monday of a week with no events in it and load the game as your mayor on that date.
I hope this helps and isn't too confusing... it sounds complicated but it's really easy and it's the method I use when I want to move out only specific villagers without losing others. I find that it's very safe because you're not relying on moving gossip and you're using a new character to check ahead, plus it's fast, because you're not going through small spans of time.

If you have any questions, just ask!
Don't forget to enable the beautiful town ordinance to keep your flowers from dying. You'll also get some weeds and bed hair from this method but it's worth it.