backing up mail in apple mail

Give Mail Backup X a try.
I have been using it for past few months. Offers Automatic, Incremental backup.
It backs up all my On my Mac mails with the currently arriving emails correctly.
It backs up Outlook and Thunderbird apart from Apple Mail.

I don't have the exact configuration as you do have, so I can't guarantee if that would work for you as well.

However, you can give a free trial to ensure this is what you exactly need(The tool offers up to 15 days of free trial).

Link for Mail Backup X tool:
http://www.mailbackupx.com/how-to-scenerios/how-to-backup-outlook-mac-mails-on-m ac.html

Thanks for the clarification. If both email accounts are IMAP accounts, then all of your email messages for both accounts are on their respective email servers ... except for any messages that you downloaded locally to your Mac and placed into email folders.

I would second rccharles advice about exporting all of your mailboxes. Another option is to use a third-party app, like MailSteward to archive all of your email mailboxes. You can then access or restore them as needed and this would make for a good backup as well. In addition, Time Machine backs up your Mail app's mailboxes as well so you should make timely backups with it if you are not already doing so.

I'd export all of the messages. I assume the same strategy will work in High Sierra.

Mail for macOS Sierra: Import or export mailboxes

more info with:

https://www.google.com/search?q=export+message+macos+mail&ie=utf-8&oe=utf-8

Time Machine backs up all email messages.

Time Machine back up all files not email messages. You would be left with the problem of figuring out how to restore your email messages. With imap the messages are left on the server until you want to look at a message.

You could rely on the imap server to keep a copy of all your messages but it never hurts to have your own backup.

R

Sorry, GD=? Would that be Google Docs? ... and by Office 365 are you referring to your Microsoft account email? Regardless, how are you accessing these email accounts? For example, are you using a web browser, a dedicated app, or the Mail app on your Macs?

What backup method do you use currently? Time Machine backs up all email messages.