You can take a backup on your old computer in your old version.  Then transfer that *.cab file to the new computer and after you have installed and registered the software, start it up and select Restore.  Select the *.cab file and let the program restore and upgrade the file.

If your old file is from prior to 2015, you may need to install the 2015 trial version of the program to upgrade to 2015 before you upgrade to 2022.  If you do, then you just need to restore with the 2015 version then when it converts the file to 2015, open the file in the 2022 version and it will trigger the upgrade automatically.

Upgrades are one-way only.  Once upgraded, you can't go back so don't get rid of your old backups and old software for a while.