If you will not migrate to new Operation System (eg. Windows - Linux) there is no problem for you. You may create a Firefox account (or use existing) and log in to it on new desktop for synchronization in bookmarks, settings and other stuff.

https://support.mozilla.org/en-US/kb/how-do-i-set-sync-my-computer Sync your bookmarks, history, passwords, add-ons and open tabs with another copy of Firefox. Learn how to set up Sync.

https://support.mozilla.org/en-US/kb/how-do-i-choose-what-types-information-sync-firefox We'll show you how to choose which types of information (bookmarks, history, add-ons, and passwords) to share across all your devices through Firefox Sync.

https://support.mozilla.org/en-US/kb/firefox-sync-troubleshooting-and-tips

https://support.mozilla.org/en-US/kb/sync-firefox-bookmarks-and-browsing-history-iOS Sync your passwords, history, tabs and other browsing information across your iOS, Android and desktop computers with Firefox Accounts.

Please do keep a backup copy of the old profile folder in case something goes wrong:

Back up and restore information in Firefox profiles