Mike,
I believe you are referring to the ArcGIS License Manager. If so, moving licenses from one machine to another should not be too difficult. Here's the process:
1. Deauthorize all licenses from the current license manager on your first machine. This will return all licenses back to the licensing pool to be authorize later.
2. Locate your provisioning file if you have one. If not, go to MyEsri to download or create a new provisioning file. It's basically a text file containing your information and selection of licenses to authorize.
3. Install the ArcGIS License Manager on the new system and authorize licenses using the provisioning file in the previous step.
ArcGIS License Manager 2018.0 quick start guide—License Manager Guide | ArcGIS Desktop
If you're unfamiliar with any of the steps above, our help documentation can walk you through the process. Although there are quite a bit of information to go through. If documentation is too intimidating, I suggest your contact Esri Technical Support. One of the analyst can walk you through the process.
