Restore Time Machine Backup on an older OS
I did the same thing by using a USB installer and migrated data from the Time Machine backup during the install (it's one of the choices during the install). In my experience the data (minus the OS) should come back to Mojave but it may not be visible in the list of backups to choose from when choosing a backup source during the install. To explain further:

When I installed Catalina I chose to migrate data from the Time Machine backup of Mojave which was successful. In the brief period of time I had Catalina it made maybe 1 or 2 Time Machine backups. When I chose to erase the SSD, and install Mojave again, Catalina appeared to have changed the file extension of the backup from .sparsebundle to .backupbundle . This stopped Time Machine on Mojave from being able to show me the backup to select (imagine my heart rate). So, erased again, installed Catalina, and after browsing the Time Machine backup in Finder, I could see the old backup there, changed the file extension from .backupbundle to .sparsebundle , erased the SSD again, installed Mojave and presto, the previous Mojave backup was visble again. The data migrated successfully for the most part (except my Contacts which is a whole other story).