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 flow from the Partner to the customer. The Partner must separately request a refund from you. If the Partner is using the Catalog API, they create a refund request using their refunds endpoint.
Brand processing refunds
If you are processing the refund, issue it as you would for an order that came through your site. The refund flows from you to the Partner to the customer.