Both the Partner and the Brand can process a refund for the customer. Catalog recommends discussing with your Partners how you prefer these situations to be handled.
Partner processing refunds
If the Partner is hosted on Shopify, refunds will flow from the Partner to the customer. They must separately request a refund from you. If their Partner is using Catalog API, then they will create a refund request using their refunds endpoint.
Brand processing refunds
If you will be processing the refund, you will issue this as you would for an order that came through your site. The refund will flow from you, to the Partner, to the customer.