This is a straightforward “how-to” article to help you when migrating mailboxes from your on-prem Exchange server to Office 365 in a Hybrid configuration. It is assumed that you already have the hybrid setup configured with either Exchange 2010 S3 or 2013.
Open the Office 365 portal select Exchange from the management menu
Select migration under recipients
On the migration page, select Migrate to Exchange Online
The dialog window opens, select Remote move migration, next
Select the + icon to choose on-prem mailboxes that you want to move, next
If this field does not auto-populate, enter the external FQDN of the Exchange server, next
Name the batch anything you want, next
Select options for starting and ending the batch, next, window closes, reverts to portal
If you selected to manually start the batch, you will have to select the Start icon
You may have to refresh to see the current status of the batch, when it is complete you can view a report (including errors) by selecting the view details link
Once this is complete you will have to change the target email server on the user’s phone to outlook.office365.com.