By default, the iPhone does not save Drafts, Sent and Deleted(Trash) emails to the server. If you want those emails to be available to all devices that check the server, there are a couple of steps you'll need to do in order to enable that.

1. Open "Settings" on your iOS device
2. Scroll down the menu and choose "Mail, Contacts, Calendars"
3. Select the account you wish to modify
4. In the next screen, select your account again
5. Choose Advanced at the bottom of the next screen
6. Select each folder and choose the corresponding folder on the server to save emails to
7. Once you have completed the previous step for each folder, tap "Account" in the upper left of the screen
8. Back in your Account screen, tap "Done" in the upper right of the screen

Now your emails will be saved to the server instead of just to your iPhone.