Voiding is when a villager is sent to the void; moving out of town with no place specifically to go. If you connect with someone via wi-fi, local play, or streetpass, they have the possibility of picking up a villager from your void and vice versa. I'm not sure how many can be stored in the void at once, but I believe it's only one. They'll only move in if there are only 9 villagers in town.
There are a few cycling methods you can find online. There are ones specifically for people who want to keep some villagers but not all, and some that will move out everyone one by one.
This guide is an example of one that will allow you to choose who you keep, and
this one will get everyone out. Choosing who you keep will make it take longer to cycle though. Basically, their town is just for moving in and out villagers. Most people who do this have more than one town and set up their second town as a cycling one.
Trading villagers is simpler. If someone is "in boxes" aka it is their moving-out day, someone can visit your town and talk to that villager to convince them to move to their town. This can only be done if there are only 9 villagers in the town the villager is being invited to and no one has their house plot down. After they've been convinced, if you talk to them before the person leaves, they'll tell you which town they're moving to. After the person leaves, if you go the villagers house, it'll say "Moved Out." when you interact with the door as if you were trying to go inside, and the next day they'll be gone. It take 1-3 days for a house plot to show up in the new town and 2-4 days to fully move in.
