How to copy/export emails from old iPhone

If at least one of the Mail Accounts available on both phones you can

• create a mail folder in this account
• copy all mails from the shutdown account into this folder (on the old phone)
• wait a while to have the phone sync the mail folder back to the Server
• access the Mail folder from your new phone

Turns out I was completely wrong in assuming that you could Backup non-iCloud emails either on to iCloud or to a computer.

Even though Apple docs say:

Some information is not included in an iCloud backup but can be added to iCloud and shared across multiple device, such as Mail, Health data, call history and files you store in iCloud Drive.

Emails from your device cannot be copied to iCloud Drive even if you enable the Mail app to use iCloud Drive.

• When you enable 'Mail' in iCloud Settings, it only enables your iCloud email account to use it with the Mail app.

• When you enable 'Mail' in iCloud Drive Settings, it only backups and saves all your email accounts settings and 'Mail' app preferences.

Both options do not save a copy of all the emails in your device to iCloud. (You can however save email attachments manually to iCloud drive).

Another option is to create Backups to a computer. But even choosing to opt to backup "all data" on your device to your computer, using iTunes -

does not copy your emails to the backup - only the email account settings and Mail app preferences are saved in the backup.

So the only way to copy and transfer non-iCloud emails that you no longer have access to, is to transfer the emails to another active IMAP enabled email account on Mail, as described in this answer, and then download it again on to the new device.