In the Animal Crossing series, villagers will move out after staying in a town for some time, though some, but rarely, will move out only a few days after the start of the game. In the original Animal Forest up to Animal Crossing, villagers leave without warning, appearing in town one day, gone the next. Some neighbors are easier to convince to stay than others, while other neighbors can be tough to get to leave.
If a neighbor asks to move away, the player has the choice of saying either "Don't go!" or "Take care!/Good luck!" Different animals have different responses. If an animal has been thinking on whether to move or not for quite a few days (even a week), he or she will move away. It may take the villager anywhere from almost a day to a few days to pack and leave. When a villager has packed all of their belongings into boxes in their house, they are colloquially referred to as being in boxes. Once a villager moves out, his or her house vanishes and leaves behind a patch of dirt and a signpost. A couple days after a move, a new villager will move in on top of a signpost somewhere in the town. Signposts were removed in Animal Crossing: New Leaf, in which a villager can move to any place in the town.
In Animal Crossing: Wild World, the boxes are marked with a Tom Nook leaf, suggesting Tom Nook moves people out as well as in. Villagers can move from a friend's town if the player has them on their Friend Roster. In Animal Crossing: New Leaf, villagers can visit Main Street under the guise of 'shopping' or 'visiting'. If the villager that moved was the only animal in town to have a certain personality then another animal of the same personality will move in- this ensures that there is a balance of characters in the game. Villagers may move to another player's town provided there has been some form of contact between the two towns some time previously. The Player can also move out by going at the Civic Center and selecting "Moving".
When a villager is planning on moving, the best way to get them to stay is through talking to them, then writing them a letter and attaching a present. In Animal Crossing: City Folk, sometimes villagers that have lived in the player's town can be seen in the city, but they will not recognize the player. In Animal Crossing: New Leaf, sometimes villagers from other towns that have been StreetPassed will move into the town. The villager will talk or mention its former hometown occasionally.
Animals move in at the beginning of the game, one at a time, or until one of the other neighbors moves out. In Animal Crossing: Wild World, the town starts out with three neighbors, and five minimum (minimum because it is possible a villager will move out within a few days of beginning the game) will move in. Villagers will move in on top of a random signpost. If an animal is moving in, then they will not leave their house until they have unpacked. The first time the player talks to them, they will introduce themselves and will be very pleased to meet them. The second time the player talks to them, they will tell the player to come back the following day as they are too busy unpacking. Lazy villagers may whine about needing to unpack their pillow for a place to sleep, snooty villagers complain about how their house is such a mess, cranky villagers get angry and yell for the player to go away, and normal villagers will feel like they need some time to move on in.