With VPN connectivity, remote desktop may not find computers by name but using the IP address works. This happens when your computer is not joined to the remote domain. Wireshark inspection of the DNS packets shows the query is for “desktop1.home” instead of “desktop1” you entered in the search box. Since the remoted DNS has no records for .home, it returns a not found response. Now it is necessary to use the full FQDN instead of just the name as you would on site. You can do that with each remote desktop session or change the default so you can just enter the machine name.
Controlled under the IPV4 settings on the Ethernet connection (not the VPN connection). Right click > select properties > select IPV4 > select properties > select advanced button > select DNS tab > select button ‘Append these DNS suffixes’ > select Add > enter the correct suffix for the remote domain.