Another Update after much more testing.
First off we know that the Group policies for computers ONLY updates on computer startup when connected to a site.
The testing we are doing is in regards to users with laptops moving between sites and how it will behave based on what users will tend to do with the laptops (just shutting the lid and leaving to another site).
We gapped the networks (network disconnected) for testing only. Knowing that the other site was not available to see what the client computer would report with the "Set l" command. This proved to us that it was cached and not correctly showing what server was really used because the other site was physically not available.
We have found that if you do not let the client computer cache the login credentials at all (registry change) then the problem mostly goes away with the "set l" command. It does not show the wrong login server when moving sites. But with that ability removed for the client computer, it will stop the login at all if there is no connection to any domain controller(not wanted).
We also found by using Wireshark and Netlogon logs, even though the client computer shows wrong logon server with the "set l" it does SEEM to use the correct DC for authentication.
We know that the users should log off and shut down before changing sites. We were trying to see what all would happen when they don't follow the rules as most user will tend to do.
At this point we feel the largest issue is that the "set l" command is not showing the correct logon server, it seems to only be reporting the last one used. We completed a test as follows. Move client computer to site 2, login and run "set l" it showed site 1 as the logon server. logged off, shut down moved to site 1, started back up, logged in and ran "set l" it then showed site 2 as the logon server. Again it was the last server it used, not the current server. Was able to recreate over and over.
All of this testing is because we went down the rabbit hole of trusting the "set l" command at first. We have this question posted on other forms and have see so many responses like "who cares unless you are seeing some sort of an issue" Our outlook is we do not want any issues that could have been avoided with some time and care before production.
Thanks to anyone who does post to this and gives any sort of suggestions. We again are testing like this because we know users don't follow directions well with the use of technology and best practices at this point.