Remapping domains after creating wordpress staging site

I think this scenario will be really common but I cannot work out how to do it with easyengine.

I set up a site for build / staging on a VPS lets call it

I build the site and show it to the client and we agree it’s all good.

What is the best way to perform migration/remapping of this site onto their domain name?

I understand the DNS aspects of it but it’s what I need to configure on the easyengine server that I do not understand.