The Migration requires that the destination is the higher level MacOS (or at least equal to the source MacOS). It does not work when trying to migrate from the higher (Monterey) to the lower (MacOS) level with Migration Assistant.
If you have a backup (Time Machine or other) made from back when you were on Big Sur, you can migrate from that backup.
If your only backups are in Monterey, you can do the "migration" manually. This could involve:
(1) copying manually all files in your Documents folder to what will be your new/restored system
(2) copying over manually the folders for Safari, Firefox, Chrome etc. browsers. As an example, for Firefox this folder is called Profiles and it is in your USER Library => Application Support folder. Not sure if this will work well for Safari since Safari is part of the MacOS. There is a Safari folder in the USER Library folder, and I believe that Safari folder has bookmarks, etc. in it.
(3) For Mail, if your email is all IMAP or GMAIL or Exchange, you can simply recreate anew the email accounts and the emails will download automatically. Mail that had been saved locally (On My Mac) might be importable from the Mail folders that are in the USER Library folder.
(4) Applications would need to be reinstalled anew. Also, reinstall printers, reestablish internet and WiFi settings. This all takes a little time but is not hard to do.
You may have good reasons to go back to Big Sur. But before you do that, you might try erasing the entire physical drive and installing Monterey from scratch and then migrating only user accounts and user files, no settings and no applications. Then see if the plain system works ok, then install applications piece meal, checking after each for stable operations. If this still does not work, you could embark on the return to Big Sur.