When Moving to another Server, or when changing the server IP address the best approach is to first change the AEA Configuration with the new IP/FQDN and let the clients check in and update. Also do not forget to rebuild your deployment package that you use for the .bat install.

To change the address, go to the group in EAC that you want to change and right click and choose properties (Or do this for the Computer Catalog if you do not have groups)

Then go to Communication and here enter the address of the server. Also be sure to check the option to try to find a fail-over server if it can not contact the EAS