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 users' inbox or other mail folders are migrated. Contacts, calendar items, and tasks can't be migrated with IMAP, but they can be by a user. You'll have to create a mailbox for each user before you migrate their email.