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