I've been with the same firm for 12 years. Back in the day we didn't have a license server. We installed our first one when Autodesk made us pick stand alone or license. That was about 10 ears ago. So that same server has been chugging along for 10 years and I got a little nervous about its life expectancy. I did some research on upgrading a license server and came up with... well about nothing. I pulled it off on faith and I'll share what I learned in case others are about to do what I just did.
First pick a new machine and get a new LIC file from Autodesk. That is all documented well in other places. There is also plenty of documentation on getting it up and running. What doesn't seem to be out there is what happens next. Two license servers running at the same time didn't give me the opportunity to peel off one machine at a time to the new server until I was done. I had to come in on a Sunday and shut the old server off and hope for the best. The new server did not kick in and pick up sending out licenses. I had to add an environmental variable to each machine telling it to look to the new server for a license. I was sad because it still didn't work. Turns out the new servers firewall was block the PCs from asking for licenses.
Once I got one to work, I just needed to go to each PC and redirect to the new server. A friend of mine had a similar experience, but his users just had to type the new servers name in a dialog his Autodesk products would bring up.
Think over the scope of what is in front of you and get help if needed. I had 3 satellite offices to cover along with the headquarters building. And lastly be prepared to go backwards if things don't work out.
While you are at it consider this. I picked up two new servers and got LIC files for both. If anything happens to my new server, its replacement is already broadcasting the new licenses. I hope I never need it, but if I do... down time should be short, and I'll be a hero.