Mystery of the unmovable mail (and how to sync iOS email folders across all your devices)
Moving to your Mac, you discover you can delete the message without a hitch and that the message you delete from your Mac also disappears from your iOS device. But none of your iOS devices will let you delete the file.
You also note that any draft messages you create on an iOS device can only be found on the device they were created on. Not on any other iOS devices and not on your Macs.
Finally, you can't locate any of the messages you've sent from your iOS devices when you look for them on your Mac.
Why
You're using iOS Mail's default settings, which use local storage on your iOS device instead of your mail server for storing drafts, sent, deleted, and archived mail. But, never fear, we can fix this in a flash.
Where's the Mail Stored
When you use IMAP mail on your Mac (and, trust me, you should be using IMAP) the default location for storing drafts, deleted messages, sent mail, and archives is on your mail server. You can see where your Mac stores these files by clicking the Mailboxes link in Mail's toolbar. When you do, you should see something similar to what's in this screenshot:
While it's not obvious from this screenshot or from looking at these folders on your Mac, everything here is a mirror image of the same folders on your mail server and they're synchronized with the same folders and on any other Macs you have that use the same mail account. Delete a message on one device and it's immediately deleted on all your other devices. Create a draft on your Mac and that draft will show up on all your other devices too. (And that, my friends, is the beauty of IMAP.)
Here's the important point: whatever you do on your Mac will be reflected on your iOS devices too. Here's how to fix mail on your iOS device so that it works like mail on your Mac.
You will need to repeat this process for each of the Sent, Deleted, and Archived mailboxes on all of your iOS devices. You will also need to run through this process for all of your email accounts. But once you do, all of your devices will share the same mailboxes for all your mail.