Perform a mailbox migration request between Exchange 2010 and Exchange Online

Firstly ensure that the admin user is within the 'Recipient Management' (and also 'Organizational Management') role group:

Add-RoleGroupMember "Recipient Management" -Member admin1

Verify with:
We should also ensure that our MRS proxy service is enabled with EWS by issuing:

Get-WebServicesVirtualDirectory | Set-WebServicesVirtualDirectory -MRSProxyEnabled $true

Now lets connect to our Exchange Online tenant:

$UserCredential = Get-Credential
$Session = New-PSSession -ConfigurationName Microsoft.Exchange -ConnectionUri https://outlook.office365.com/powershell-liveid/ -Credential $UserCredential -Authentication Basic -AllowRedirection
Import-PSSession $Session

and then from the shell we can run something like:

$RemoteCredential = Get-Credential

New-MoveRequest -Identity <user-to-migrate> -Remote -RemoteHostName <your-on-premis-exchange-server> -TargetDeliveryDomain <destination-domain> -RemoteCredential $RemoteCredential

** If you get any access denied messages while performing the move request ensure that the account you are performing the operation with ($RemoteCredentaial) is inheriting security permissions from parent objects. See here for more information: https://support.microsoft.com/en-us/kb/2975731 **

We can get the progress of our move request with something like:

Get-MoveRequestStatistics <username>