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Microsoft has a page in its Exchange support section on mailbox moves in Exchange 2013. There are too many steps to explain here, but it all starts with a "move request." You will need to be assigned permissions before you can move mailboxes to another server.
A "batch moves" function included in Exchange 2013 enables the moving of multiple mailboxes in large batches. To move mailboxes from 2007 to 2013, you'll need to use the Exchange 2013 admin center and the Exchange Management Shell, according to Microsoft.