So, say I have a villager you want and you came over the day that villager was packed up and "in boxes". You'll know they're "in boxes" because inside their house, there are boxes everywhere.
Anyways, you come over, and visit that villager. During the conversation, you ask the villager to move to your town. At that point, the deed is done.
If the town mayor visits that villagers house, they get a "moved out" message. The next day, the house is gone from that town, and the move in process begins for your town.
Best to post when your villager is ready to go or "in boxes".