Cycling means moving villagers in and out of your town.
So cyclers have towns where they just have villagers move in and let people adopt them when they're ready to move out.
When you need to cycle 16 to get a villager back, it means you need to have 16 other villagers move out after that villager to be able to have them in your town again.
there's no set way. I usually just introduce myself to new move-ins & never talk to them again until they ping to leave. Villagers leave my cycle town every 7-10 days.
Villagers will randomly decide to move out every 2-3 days. After that, there's 4 days where it's possible to stop them, as they go into boxes on the 5th day and it's not possible to stop them at that point.
When they're in boxes, somebody can come into your town and adopt them. If they aren't adopted, they go into your void. Their house will be removed the day after they go into boxes, as that's when they move out.