Hi,
As there are VPN connected between two sites, on network level, both sites are on same network, so, make sure that necessary ports (such as AD and other services communication) are opened via VPN network. Then, move computers to new site, establish VPN to old site first, other services should be still working.
Service overview and network port requirements for Windows:
https://support.microsoft.com/en-us/help/832017/service-overview-and-network-port-requirements-for-windows