Apple Mail stuck on moving messages (1 of 19).

I found a solution to this issue on macOS 10.15 Catalina. To fix the issue, the Apple Mail's database must be rebuilt. Once I completed the referenced steps, Apple Mail no longer had issues moving emails to folders.

https://c-command.com/spamsieve/help/how-can-i-rebuild-apple

Hi Rudolf Hacker,

Thanks for reaching out here in the Apple Support Communities. We see that you're having an issue with your mail not coming to your iMac, and being stuck on a "moving messages" step. Also, your fans are running fast and your iMac is heating up.

You've done good troubleshooting by restarting, and we have some other steps for you to try.

For the fans and heating issue, we recommend that you reset the system management controller (SMC). This reset won't affect any files on your Mac.

1. Shut down your Mac, then unplug the power cord.
2. Wait 15 seconds, then plug the power cord back in.
3. Wait 5 seconds, then press the power button to turn on your Mac.

Before doing any troubleshooting with your email account, please make sure to back up your iMac so that your information will be safe. This article explains how to do that: How to back up your Mac - Official Apple Support

Once you have a backup, what you try will be based on what kind of account you have.

If it's an iCloud or other IMAP account, you can remove the account, and then add it back in again.

Add or remove email accounts in Mail on Mac - Apple Support

If you are using an Exchange account, you'll want to check with your Exchange server support to see if anything is causing the messages to not be moving to your Mac.

Take Care.