There is no way to transfer messages between devices. It's assumed that your mail messages are stored "in the cloud" (on mail servers) and you can always have the app sync them from there.
This is of course only possible with IMAP and EWS, which allow the uploading of sent messages (POP3 does not).
As an alternative (for POP3), you can configure "automatic BCC" on your outgoing messages... it doesn't even have to be the same mail accounts (can be anything). This is under each account's Options and Folders -> Composing and sending (long press the account within the account list to get there).