Remove the Exchange 2010 default Mailbox store

The First database contains the SystemMailboxes which are the Arbitration mailbox(es).

We have to move these system mailboxes to another database before of remove the Default Database.

1. Find the Arbitration Mailbox using the Exchange MAnagement Shell (EMS):

Get-Mailbox | Where {$_.Name -like "Discovery*" } | ft Name

You might need to move the Federation Mailbox also...

Get-Mailbox -Arbitration | ft Name

2. Now create a new move request in order to move the system mailboxes to another mailbox database:

New-MoveRequest -Identity "SystemMailbox{yourGUID}" -TargetDatabase "dbxxxx" 

New-MoveRequest -Identity "DiscoverySearchMailbox{yourGUID}" -TargetDatabase "dbxxxx"  

New-MoveRequest -Identity "FederatedMail{yourGUID}" -TargetDatabase "dbxxxx"

3. Verify whether the move request completed:

Get-MoveRequest

4. Now remove the move request

Remove-MoveRequest -Identity "the name of the request like Microsoft Exchange Approval Assistant"

Or clear the lot...

Get-MoveRequest | Remove-MoveRequest 

Edit: For Exchange 2016 you can use these commands:

Get-Mailbox -Arbitration | New-MoveRequest -TargetDatabase "dbxxxxxxx"
Get-Mailbox | Where {$_.Name -like "Discovery*" } | New-MoveRequest -TargetDatabase "dbxxxxxxx"
Get-Mailbox -Database "Mailbox Database Old" -AuditLog | New-MoveRequest -TargetDatabase "dbxxxxxxx"
Get-Mailbox -Database "Mailbox Database Old" | New-MoveRequest -TargetDatabase "dbxxxxxxx"