In the original Animal Crossing, you can't stop animals from moving. In Wild World, when you go up to a player in their house who has boxes, go talk to them. You will be given the usual options of stopping talk to them and continuing the conversation. Pick Continue. They will tell you sadly that they are moving, and then you will have to options: The top one being to beg them to move, and the bottom being letting them go. Choose to let them stay. Beg them a few times, and they will stay.