If your organization is like a lot of them out there, you have mailbox users without the email address policy applied. It's a pretty common practice to work around name changes, users with similar names or middle initials, or one-off primary SMTP addresses.
If that's the case, then you probably have received this error when trying to move a mailbox:
"Target mailbox doesn't have an SMTP proxy matching '<domain>.mail.onmicrosoft.com'"
We detail this message and the cause in https://support.microsoft.com/en-us/kb/2939340, but how to fix it for a bunch of people? Fortunately, there's a script for that.
Download this script and run it from the Exchange Management Console. You can specify a -Searchbase to scope it to a particular OU as well as target all users without the email address policy applied or only a specific user.