Completing a Quick Ship Order will depend on the delivery method you've chosen for the Quick Ship Order.

Select Drop Ship from the delivery method drop down

Like with the previous 2 options to quick ship, you can choose to apply to all or select each one individually.
Once the option has been chosen and applied to each item, select save on the bottom right

Once Saved, your order will change its status to "In Progress" and automatically tag it with the name of the third party shipper

It will also show you all the items shipped for that order, immediately after it will show them as "Requested" in an orange box

Once the order is received by the vendor the item status will change to "accepted" in a blue box

Once the items ship it will change to "Shipped" in a blue box and a new box that says Track It will appear for you to click on and track the package. 

This will also cause the system to create the invoice for billing and the order will change to completed.