Creating in a subdomain

I am creating a site for a client and he wanted to keep his site live while I was building the new site. So I am building in a subdomain. I have a few things left that I need to do before transferring the site from his hosting service to mine but, admittedly being new to this arena of design, I ask... can I build a child theme now or should I wait until the site is transferred to the permanent hosting position and under its own domain name instead of my subdomain. I'm not sure if child themes change when transferred to new host with its proper domain/url name. Thanks for any help - Jeff