Under Account Setup you can set the archive or flag command to move to a folder which you can specify. This will move it on your mail server and it will create the folder if not there.
If you don't see the new folder or don't see the mail flagged in your desktop mail client, try logging into your mail account using the web version.
If it is reflected correctly in your web version but not your desktop, make sure you are connecting to your server using IMAP and not POP. You need to use IMAP to see changes from another mail app.
If it is not correct in your web version either, check your seever settings under Account Setup to make sure it is not set as POP on Speaking Email. If this does help, email us and we can review logs.