Apple Mail stuck on endless "Moving Messages" — How to fix?

Hello GroovySteve! Thank you for participating in the Apple Support Communities.

We understand that Mail appears to be stuck on moving messages in Mail on your Mac for your Gmail account, while the same account isn’t experiencing this issue on your iPhone. We'd like to look into this with you to see how we can help. 

To begin, does restarting your Mac have any effect? Log out, sleep, wake, restart, or shut down your Mac

If the issue persists, let’s try starting your Mac in safe mode. This can help determine if this issue is related to software loading when your Mac starts up: Start up your Mac in safe mode

Make it a great day!

Hello GroovySteve! Does the issue persist if you try connecting to a different network?

Are there any updates available for your Mac, or is up to date with the latest macOS available for your Mac? Update macOS on Mac

If the issue persists, let's run First Aid on your disk to see if that helps: How to repair a Mac disk with Disk Utility

Cheers!

If you are using GMAIL IMAP (not GMAIL POP) then you can try removing that GMAIL account, reboot into Safe Mode, then boot normally and add that GMAIL account back. With GMAIL IMAP, when you re-add the account back it should synchronize everything with the GMAIL server immediately (like your phone does). Hopefully it will behave after removing and reinstalling your account.