What do I have to do on Shopify when there are cancellations, refunds or returns on the marketplaces?

Cancellations

Cancellations will automatically be handled by the app on all platforms

Refunds and Returns

For Lazada and TikTok the app will handle the refund and restock for you once the return has been processed on Lazada and they've sent the app a notification about it.

For Shopee, cancellations will work in the same manner, but there is a slight difference for returns. In this regard, you will need to refund the order manually because Shopee made it ambiguous for us to know whether the order has been "completed" or "returned". IE they both just show up as "completed"

For more info, please have a look here: Why are "returned" Shopee orders not restocked on Shopify?

As to the actions required, since the imported order is a Shopify order, for imported Shopee orders that were returned (not cancelled), you can proceed to refund them like a normal Shopify order (https://help.shopify.com/en/manual/fulfillment/managing-orders/refunding-orders)

To manage expectations, please note that the updates you do on the imported Shopify orders will not affect the actual order on the marketplaces.