To perform this action will require booting from a Bootable Installer.
This will have to be performed from a Qualifying Computer to run the version of macOS to be made on the Bootable Installer.
Example : Bootable Installer of Big Sur would have to be done on a computer that Qualifies to run Big Sur.
If the computer being used to download the older version of macOS Does Not Qualify to install the older versions of macOS - it can not be done from this computer - period.
Apple has a Licence of Usage where it outlines basically, one can not downgrade to an older version of macOS that the version it came installed with as New.
To make it more clear. If the computer being used came with - say Catalina macOS 10.15 as New and one what's to download ( even if only to make a bootable Installer ) let us say Mojave macOS 10.14 or High Sierra macOS 10.13 - this computer is disqualified to be used for this purpose.