No. When you refund a transferred Etsy order in Shopify, Shuttle does not send that refund to Etsy. Your customer only gets their money back when you refund the order on Etsy.
Why a Shopify refund does not pay your customer back
A Shopify refund on a transferred Etsy order returns no money, because your customer paid on Etsy, not through Shopify. The payment you see on the order in Shopify is a record that Shuttle adds so the order totals match.
No money came in through Shopify for a transferred order. So refunding it in Shopify only marks it as refunded, and nothing goes back to your customer.
How to refund an Etsy order
Every refund for an Etsy order has to start on Etsy, because Etsy does not let outside apps such as Shuttle issue refunds.
In Etsy, open the order and issue the refund there.
Leave the Shopify order as it is. To show the refund on it, update the order from the Orders page in Shuttle.
How Etsy refunds show up in Shopify
When automatic order transfers are on, Shuttle copies an Etsy cancellation to Shopify for you. A refund without a cancellation is not copied automatically, so you update that order from Shuttle yourself.
When you cancel an Etsy order
When automatic order transfers are on and you cancel an Etsy order, Shuttle marks the matching Shopify order as refunded. Where Shopify allows it, Shuttle also cancels the Shopify order.
To transfer orders automatically, you need the Growth plan or higher.
When you refund an Etsy order without cancelling it
Shuttle does not copy a refund without a cancellation to Shopify automatically. To record the refund on an order that is already in Shopify, update the order from Shuttle:
In Shuttle, open the Orders page and click Transfer orders.
Choose Transfer or update orders.
Choose Only update existing, pick the orders to process, and start the run.
Shuttle then marks the matching Shopify order as refunded, for the full amount or for the part you refunded.