You can use the Internet Message Access Protocol (IMAP) to migrate user email from hosted Exchange and other email systems ...

You can use the Internet Message Access Protocol (IMAP) to migrate user email from hosted Exchange and other email systems that support IMAP migration. Only the items in the user's inbox or other mail folders are migrated. Contacts, calendar items, and tasks can't be migrated with IMAP, but they can be imported by the user. You'll have to create a mailbox for each user before you migrate their email.