***USE THIS AT YOUR OWN RISK***
Make sure you are on speaking terms with every villager. Play until someone pings to move.
*If it's someone you want to keep, tell them to stay and save. TT back 4 days. Walk in front of every one who is outside for a few seconds without talking to anyone. If no one pings you TT forward 4 days. Walk in front of everyone outside for a few seconds each without talking to anyone. About half to three-quarters of the time your mover will ping you. If not, talk to everyone a bunch of times til either someone says so-and-so is moving or til
two villagers mention "rumors" found here.
http://www.belltreeforums.com/showthread.php?145364-Villager-Moving-Rumors-(guide)
It's important that one of these two things occurs because rarely you will have movers two days in a row and that jump back 4 days after telling someone to stay could be the day someone thinks of moving.(since any length of jump backwards counts as one day forward) If someone did decide to move that day you are on day 4 of having a mover which means they would be boxed up the next day. Movers box up 5 days after they first think of moving.
If two villagers mention rumors then TT 1 day forward and walk by everyone again til you find mover. If no one pings, this is the first day you can potentially have someone thinking of moving so you are safe to look for your mover like this for a few days.
For example. Say I jumped back in time til April 2nd after someone pinged and I told them to stay.
I walked around on April 2nd and no one pinged so I TTed forward til April 6th. Still no one pinged and two people mentioned the rumors found in the guide. I can safely TT to April 7th. The new date where someone can be boxed up is the 12th. (5 days past April 7th) If I had gambled and moved forward from the 6th without two villagers talking about rumors I could have someone boxed up on THAT DAY(the 7th) or the 11th at the earliest. (5 days past April 6th.) I tend not to gamble with my villagers so I walk around everyone outside until someone pings each day...making sure it doesn't get to be 5 days past the safe zone.
---It's worth noting here: The FIRST day someone thinks of moving is always on a day you load your mayor. So I know if I don't have anyone thinking of moving on April 2nd, the first day someone can think of moving is the day I skipped forward to, April 6th. No one can start thinking of moving on the days I skipped (april 3rd-5th) If someone is thinking of moving on April 2nd, their boxes day will be on the 7th even though I didn't load my mayor 5 times. The days still count because someone was already thinking of moving.
If someone you want to stay pings. Tell them to stay and repeat the above steps.
*If someone you want to move pings either tell them to move and save OR just quit without saving and TT to the day they gave you as their move date. From this day you have to play until someone new pings. I always try to invite campers or pick up avs from cyclers so I'm not stuck with eight villagers, waiting for someone to move in. (Also, this will prevent you getting someone you want as a random and them moving into a bad spot.) When you have someone move out, you've skipped forward five days so you need to make sure you are speaking terms with your villagers in the next few days. When your dreamies are pinging to move and you're bouncing back and forth 4 days you will stay on speaking terms with your villagers.
Umm, I'm really bad at explaining things but I've done the 16 villager cycle a few times and TTed people out MANY times without losing anyone so thought I'd tell you how I do it. It really just takes knowing when you might have a mover and keep tracking of that so you know it's safe to move forward or not. Good luck with your cycling!
Edit: I might edit this more for clarity...if there's any hope of me making this understandable.