So what's your method? I gave it a shot, TT two days ahead, saving, and then TT back to the present, but I found it was inconsistent whether someone would move or not (with 9 or 10 villagers).
I'll just post what I did here ^^
1. Talk to every villager you can find whether they're outside or inside. Sometimes they may tell you that someone is moving but other times not. This is important. It helps build relationships with new villagers so they ping you when they move since sometimes if you ignore new villagers they won't ping.
If there's a rumor go find that villager and if they ping, then tell them whether to stay or go. If they don't ping, save and quit and then just start the day again.
2. If there's no rumor then run around and wait for a ping. You need to stand in front of every villager outside for a few seconds to it registers it completely since sometimes it doesn't for a second or two. Now there is two options
3a. If there is a mover and you say stay then go 3 days ahead and repeat 1 and 2
3b. If there is no ping, go ONE day ahead and then wait for a ping. If there is still no ping one more day. If there is still no ping then save and quit. Go to your 3ds clock setting and change the date to a week ahead. Next start a new character, go through the steps and if a villager has left and their house is gone (look on map when the station guy gives it to you), quit the game, go to clock and change it back to the present date. The go back with your main character and stop or tell the villager to move. If all hoises are still there an no one has left, then just quit out, go back to the present date on the clock, and then it's safe to TT 2 days ahead. By then someone will most likely ping you.
*at any time do not go more than 3 days. 3 days is only if you just told someone to stay*
Also keep a list, it helps. If you've already moved out a certain amount of villagers before you have decided to cycle 16 then try to remember who and make a list. I already had 9 for Julian and 8 for Lolly so I was half way there.
Be very careful if you don't want to lose villagers. I know it takes long to do the 1 week advance and see if anyone moves but it's much better than losing your favorite villager. If you don't feel safe TTing another day because no one has pinged you then don't, just go ahead a week and create a new save. It's not like you're keeping the save file either way.
I stayed writhing the same month to be safe as I didn't want to go so far ahead and have to go back to the date.
I recommend using the campsite to get villagers faster or taking the, from cycling threads, giveaways, and other places. As long as no one else is interest end then it should be fine as you're just going to TT them out anyway. It makes it go faster
If your town is landscaped, then don't worry about fixing it until after you get the villager(s) you are cycling back for. I honestly me hated what happened but I fixed my town up today and it's doing just fine. There's not any major changes I had to do. Just plant new trees, re arrange paths, and add more flowers. That's it. It took maybe 30 mins after both my villagers were back in my main
Be patient. It take awhile to get that many villagers out but in my opinion it's worth it.
Sorry for the novel.....