deleting email on all devices at once

iCloud email should be synchronising across all your devices, provided they are all signed into the same iCloud account. Is this not happening?

Apple has some suggestions to try in If you can’t send or receive emails on your Mac to resolve sync problems.

If you are using iCloud mail, or any other IMAP email service - which includes most online mail - then your email will synchronise between all your devices. If it isn't syncing it may be possible to turn non this feature. Which email service are you using?

That is what David was trying to explain about IMAP, if you delete an eMail on one device it should delete it on all, if you read an eMail on one it should show read on all, etc.

On Mac, in Mail>Preferences>Accounts, does it say POP or IMAP under the account name?

You were provided incorrect information or you did not understand what you were told. It does not matter what email provider/service you use. IMAP is IMAP across all email servers/providers.

BDAqua provided you the correct information.

Perhaps you nave moved your emails to a location on your device that is not synchronized with email.

Hmmm, I have several non-Apple eMail addies and IMAP deletes on all devices for me!???

This advice supposedly from Apple is not correct.

You also said you were using iCloud mail and also said you were using 'a third party email'.

David is correct.

Good luck my friend! :)