Not automatically. There is technically only one person responsible for all the charges that will be assigned to one back number. You can remove charges from one person and add charges to a different person manually. Use Maintain Charges from the Financial menu to add charges to a person that isn’t connected to a horse for that show.
From the Main menu, choose Financial and Maintain Charges. You can find any person and add charges to them for the working show. If the person isn’t connected to a horse as a responsible party already, you’ll see a message telling you that so you don’t accidently add charges to someone who you don’t want to have them. You can then add whatever charges you wish.