What is the proper way to copy backup files from Catalina time machine to Monterey time machine?

See this: https://support.apple.com/en-il/guide/mac-help/mh35732/mac

It explains how you can "inherit" your backup from an old Mac to a new Mac. After using the Migration tool, the new Mac asks you if you plan to continue using the old Time Machine disk and you can then answer "yes." My daughter did this when moving from a High Sierra (old) Mac to a new Mac.

If the existing backup was on an HFS+ drive, the continuing (inherited) Time Machine backup can continue to be made to that same HFS+ drive. I now have two Time Machine drives for my Mac running Monterey -- one, a new one, had to be APFS formatted, but the old (inherited) one is still on an HFS+ drive. This is the only exception to the requirement that an APFS drive be used for Time Machine under Monterey.

In my experience, Time Machine does not work in this manner across releases. That said, you can use the Migration Assistant to transfer data from Catalina to Monterey. Recommended best practice is to only transfer user files with the Migration Assistant. Third party applications will need to be reinstalled from their original sources. Also, system settings will have to be explicitly reconfigured (this would have be needed in any case due to (e.g security) differences between the two OS releases).

New versions of macOS (since Mojave, I believe) have introduced AFPS volume. You may want to consider attaching another storage device and adding an APFS Container (to hold APFS volumes). As of Big Sur, Time Machine can also use a volume in an APFS container.

- Pie Lover