You have to time travel to cycle, as that's pretty much what it is - moving villagers in and out at a decent pace. As oppose to waiting for them to move out on their own meaning you would only gets new villager every week or so.
The method I use is to build the campsite and keep the town at 10 villagers as much as possible, since they move faster when the town is full. If any of those villagers are desirable, I tt 4 days forward, check for a mover, tt 4 days back, check again. Very rarely I won't have a mover by this time. If I don't have a mover, I go another day forward. It's a really effective method though, you can get several movers in an hour doing this.
Just make sure to talk to all the villagers you would like to keep track of (ones you want to sell or who have been lurked for) before you start to tt. As long as you talk to them just once they will be able to ping you.
If you have no desirable villagers in town just tt a week or so until you get one in the campsite or one as a random. Then follow the method above. I find I get a camping villager every 4 days so I just tend to stick to the above method generally
There's not really much other benefit to a cycling town other than helping others obtain their dream villagers and making some bells for yourself
Good luck!
Also - be sure to get your house and permit first. Nobody will start moving until you have. Which is why it doesn't hurt to build the campsite before you start
