What does message 'moving messages' mean?

Moving messages in Mail can be related to the following.

1. When you have finished writing a new email and click send, apart from the actual process of sending the email via your mail server this also involves 'moving' the email from the 'drafts' folder to the 'sent' folder.
2. When you delete and email this similarly involves actually moving the email from e.g. your Inbox to the 'trash' folder
3. When moving message(s) often multiple ones at the same time from your Inbox to another folder aka mailbox for filing purposes this of course also involves moving messages
4. If you have turned on the built-in spam filter in Mail it will optionally move messages from your Inbox to the 'Junk' folder.

Remember also that you may have setup 'rules' in Mail to automate some of these tasks - particularly number 3 above.

(A smart mailbox does not involve moving messages.)

Are you getting an Error along with that? Does it run often or continuously?

Do you have any Rules set up?

Generally it's caused by pointing some action to a null or failing folder.

https://www.webhostinghub.com/help/learn/email/mac-mail/fix-apple-mail-moving-error

Hmmm, good work, I think you're onto something, the flags & unread thing makes me think it's spotlight, since those are both Smart Folders...

How to rebuild the Spotlight index on your Mac

If searching your Mac doesn’t return expected results, rebuilding the Spotlight index might help.

1. Choose Apple menu () > System Preferences, then click Spotlight.

2. Click the Privacy tab.

3. Drag the folder or disk that you want to index again to the list of locations that Spotlight is prevented from searching. Or click the Add (+) button and select the folder or disk to add.
To add an item to the Privacy tab, you must have ownership permissions for that item. To learn about permissions, choose Help from the Finder menu bar, then search for “permissions.”

4. From the same list of locations, select the folder or disk that you just added. Then click the Remove (–) button to remove it from the list.

5. Quit System Preferences. Spotlight will reindex the contents of the folder or disk.

Manually Rebuilding Spotlight via Terminal

If the aforementioned Spotlight control panel approach doesn’t spur a reindexation of the drive, you may need to initiate it manually through the command line. Open Terminal and use the following command string to do so:

sudo mdutil -E /

This basically asks for temporary super user status, which is why Terminal may ask you for your password (it may not if you’ve used a sudo command recently or are already logged in as a super user or root. The command asks the unix tool mdutil to reindex the spotlight database for everything on the computer, including external drives, mounted disk images, etc. To re-index only for a specific drive, use the /Volumes path. For example, for an external drive named “MiniMe,” the command would look like this:

sudo mdutil -i on /

Rebuilding a drive index can take a long time, so be prepared to wait whether you do it through the System Preference panel or the command line.

Hi Brian, no problem...

if an IMAP account you'll have Mailbox Behaviors in Mail>Prefs>Accounts, check these...

You have to click on each one off those & make sure it's a valid path for that account.

You may have to use WebMail in a Browser to see if those folders exist on the Sever end.