Elias here with Shopify Support. That's a great question!
There aren't any settings that you'd need to update on the Shopify account, simply make sure that the DNS settings is pointing to Shopify with the new registrar. If you need to find our DNS records, please take a look here: Connect a third-party domain.
It's worth mentioning that domain transfers take more than a week, so there may be interruptions for your client's boutique with the domain being transferred. To avoid this, you can switch the primary domain back to the Shopify URL or another custom domain in the meantime. Once the domain transfer is complete and the DNS records are pointing to Shopify, you can then switch the primary domain back to the custom domain.
I hope this helps! If you run into any roadblocks or if you have any other questions on this topic, don't hesitate to let me know!