The File Synchronization software is used to sync files and folders between a location and another, but in no way this works for emails. Since it's release the software has used to keep two folders to appear "in mirror", meaning that content from one to appear on the other and change according to folder's content.
If you don't use IMAP or POP, then you will have to manually export those items and then import them on the other device. IMAP and POP sync emails automatically.