Technically, AfterPay is similar to the creditcard payment model.
This is how an AfterPay payment is processed:
After placing the order, an authorization request is sent to AfterPay, using the data from the checkout. The AfterPay webservice checks if the data sent is correctly formatted and complete. If so, then it does a risk check using that data. If the payment request is accepted, the total amount of the order is reserved and can be invoiced.
The API documentation has an overview of all possible responses and how to process them.
Within a few days, AfterPay sends an invoice by e-mail for the captured amount.