Ahh okay. Yes, you do need a free space in your town! When ready, you train to the other town and talk to the moving villager. They'll ask you to move in if you qualify by either never having them in your town before, or if you did have them in town and they moved out, having done the 16 villager cycle. The 16 villager cycle is moving out 16 villagers since that one left. It's to make them leave your main street. If this doesn't make sense lmk!