To bring one of your old kitsusagi over you need a Soul Caller item. These items can be bough from Cresent Curios. Attatching this item in a claim allows you to bring over one kitsusagi from the old lines onto the new lines.
Kitsusagi that have been called return in an unadorned state (IE no clothes) but could have custom hair or jewelry. This is up to the Artist's disgression. Some traits or mutations might not translate.