In general you can stop the Handle server process, copy the Handle server data directory (the one with config.dct and siteinfo.json) to a new machine, and start a new Handle server process there.
If the Handle server IP address changes, you will need to edit siteinfo.json and config.dct, and also alert CNRI (or other prefix administrator) to reflect the change in your prefix record.
If it is very important to avoid downtime, you could instead set up the new server as a mirror of the old server and have both running during a brief transition period.
Best,
Robert