Actually, my understanding of the ‘move to another town function’ is that you can only move to become a mayor of a new town. I don’t believe you can move to any already existing town, and only side characters can move — your town’s mayor can’t leave.
All of that character’s money, items etc. will be transferred during the move. Your character’s house will also be transferred at its current stage of development/customisation.
Here’s a link to more information!