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What's the situation? These are emails downloaded from a POP3 account? That's the only way I can imagine an email would be on an iPhone only.
Yes, they were automatically copied to my iPhone from a POP3 account, when I first setup the Mail app. The situation now is that the server where they came from is redundant. So I was wondering if the emails in the Mail app can be backed up or exported somehow?

Also, since they seem to be locally saved to my device, do they get copied with my "iPhone Backup" to iCloud?
 
Setup a new IMAP account. Copy all old mail from the POP3 account on the phone to the new IMAP account. There you have a backup.
Seems like a good idea if it works. It would be "move" I think. I don't see a "copy" on my iPhone. I haven't tried POP3 to IMAP. I know that I can't move emails from my Exchange to my IMAP. IMAP to IMAP is no problem.
 
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