There are a number of options: EAC (Exchange Admin Center - GUI), EMS (Powershell cmdlets) and CSV file (with names of mailboxes to be migrated).
I'll illustrate the EAC and EMS options in the paragraphs that follow.
We manage the mailbox move from the Exchange 2013 server, EX13-1 in my case. We open the EAC and go to the "recipients" section, and then to the "migration" tab:
In the section below, we click on the plus sign "+" and select the "Move to a different database" option:
We then select the users to move:
Note: if we want to use a list in CSV format, we can navigate to the location of the file further down (not shown in the screenshot above).
We click on the plus sign (previous screenshot) to display the list of users:
Once highlighted, we can add the users we want with the "Add" button at the bottom of the list:
We'll then have something like this:
We then click "OK", then "Next", and proceed to the following screen:
The status of the migration is summarized in the resulting page:
If we refresh the page a moment later, we should see the number 2 in the "Finalized" column (that was the case for me).
On the other hand, access via Outlook (now the equivalent of Outlook Anywhere) was unsuccessful for one user.
I am prompted for credentials and informed that I need to restart Outlook:
I enabled some test mailboxes on the new Exchange 2013 server and these are accessible via Outlook and OWA.
Also, four other migrated users did not encounter the problem.
Since I could not reproduce the problem, I am not sure if simply recreating the Outlook profile (rather than the entire user profile) would have resolved the problem.