Good day, cadplan45,
macOS Catalina introduced a new file system on a Mac. The entire structure of the macOS system software and how your data is stored changed. If you're not seeing the "On My Mac" folder it's likely because the locations for saving those emails are different between the two macOS versions.
This article explains the changes.
https://support.apple.com/en-us/HT210650
Try this:
- Boot up in Mojave and create a temporary folder in Mail that is saved with your mail server, not on your Mac.
- Once all the emails sync, restart and boot into Catalina. Drag the emails from the folder you just created to the "On My Mac" folder in Mail when booted to Catalina.
They won't be able to sync or mirror each other, but it should be a quick and easy way to get your "On My Mac" emails to your most macOS Catalina preference.
We hope this helps.