Mail's mailbox lists

If you set up your email account preferences (Mailbox Behaviors) to store Trash, Drafts, Junk "On My Mac," then you will see those mailboxes under ON MY MAC. Those are default mailboxes that exist on every email server and cannot be deleted from the server.

The mailboxes listed under the account header will always show all mailboxes that are on the server, regardless if they are shown somewhere else (Favorites, On My Mac).

It should NOT be transferred.

And note that if you manually drag from any Axion box to an On My Mac box the message will be removed from the server - and from other devices that may sync with it, like your iPhone; Option-drag if you want to make a COPY instead.

You should be able to delete anything under On My Mac. However, if your Mailbox Behaviors have anything set to be stored On Your Mac, they will likely be recreated or you will be prohibited from deleting them.

The recovered messages are likely things you deleted at one point, but the database index was never updated, or the raw message files were never removed, so Mail is displaying what it found.

How many mail accounts are you using with Mail? I see one named "Axion". Is this using IMAP? Exchange?

Do you have any account using POP?

For POP accounts, the mailboxes are stored under On My Mac.

You can also create other mailboxes that are stored on your mac, and of course those will also appear under the On My Mac title.

If you only have IMAP and/or Exchange accounts, and want all of your mail to be stored in the servers, you can delete all the mailboxes that appear under On My Mac --- BUT then all the messages that are now in these boxes will be deleted forever; so be careful to check that you are not deleting anything that you need.

There is no reason for On My Mac mailboxes to mimic the ones for an IMAP account, so I guess there was some sort of glitch but it appears to have been resolved somehow. It is safe to delete those empty mailboxes.