Learn more about MRS Proxy
When you use the move request cmdlets New-MigrationBatch and New-MoveRequest to move mailboxes, the move requests are processed by the following two services:
Microsoft Exchange Mailbox Replication service MRS resides on an Exchange 2013 Client Access server and is the service that moves mailboxes from the source database to the target database.
Microsoft Exchange Mailbox Replication Proxy Remote mailbox moves work over the Internet by way of the MRSProxy service.
You can start and stop the MRS services as you would any service. There's a sharing mechanism between all instances of MRS so that no two servers will attempt to perform the same move request.