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