Upon completion of their purchases, whether it be after browse abandonment reminders, emails or any other tool used to push them back to buy what they were initially looking at, an email must be sent to your ecommerce customers in which you thank them for making the aforementioned purchase. The second step would be removing your customers from the retargeting lists that you have on Facebook and/or Google as well as the cart abandonment automation so they no longer see the ad abandonment messages, emails and follow ups reminding them to buy on these channels.
** Note: Your automation will not work properly if you do not clearly specify all the required conditions.