When a multiple shipping address order comes into your store, your fulfillment process remains unchanged, but it is important to understand the process of how these orders are generated to save confusion.
When a multiple shipping address order is placed, all of the data for the order is coming into your Shopify store via a single order. To create orders per shipping address (so that you can fulfill orders on a per shipping address basis) we must therefore cancel the initial order and recreate new orders out of it per shipping address.
This is how a a multiple shipping address order going to 2 address will look in your Orders queue when received:
You can see here that order #1002 is automatically canceled, and orders #1003 and #1004 are recreated by Giftship.
If you open order #1002 you will see all of the order data, and you will see at the bottom that Giftship has canceled this order:
Even though the order has been canceled, it has not been refunded. This is the order through which money has come into your store. But the initial order should largely be ignored, as it is not to be fulfilled.
Here you will see one of the recreated orders. You can see that the gift message and delivery date information is nicely displayed, and that the shipping address information is in the usual spot. Despite a charge being present here, your customer has not been charged again - this is a recreated order marked pre-paid.
Please let us know if you have any questions.