In an Exchange cluster with a Database Availability Group (DAG), you can easily "switch over" one, more or all mailbox databases to another node.
Switchover can be done through EAC, or through the EMS (Powershell).
To switch over all mailbox databases to a server called EX2:
Move-ActiveMailboxDatabase -Server EX2
To switch over the database called DB1 to a server called EX2:
Move-ActiveMailboxDatabase DB1 -ActivateOnServer EX2
The switchover will take a few minutes to complete, depending on the load on the database that needs to be moved. During the switchover, Outlook clients can lose their connectivity up to about a minute, but in normal circumstances, they should automatically reconnect again.
To verify the switchover (e.g. for a database called DB1), run the following cmdlet:
You should see the database Status of:
You can also see the active node for the database (e.g. DB1) with this cmdlet:
The Server field will tell you the active node for the specified database. Leave the database parameter to see the active nodes (= Server) of all databases.
More information on mailbox database switchover can be found on Technet.