How to copy mail messages from offline account?

The mail server for one of my IMAP accounts was taken offline permanently a few days ago. It will be replaced with a new account with new credentials on a new server, but all of the content on the old server is gone.

Fortunately, all the messages in the account are still accessible in the Mail app on my Mac under that account.

It seems simple enough: copy all of the mail messages from the defunct account into new “On My Mac” folders; delete the defunct account; create the new account on the new server; and copy the saved messages from the “On My Mac” folders to folders in the new account, where they will propagate up to the new server and then propagate down to all my other devices.

Only it doesn’t work.

In the Mail app, I've created a new, empty folder “On My Mac”. I open one of the folders in the defunct account. I select a message, which is perfectly readable. Then I try to drag it to the new “On My Mac” folder. It fails.

The error dialog says

The message “message subject” could not be moved to the mailbox “New Mailbox — On My Mac”

The account “defunct account name” is offline.

Well, duh. That's the reason I’m trying to move the message!

How can I move the local copies of messages from an offline account?