This should be done before migrating emails across domains or email platforms to save a copy of your old emails to be imported later onto the new email server.
In Outlook Classic Desktop, go to File -> Account Settings -> Account Settings... (If you don't have this option, you are likely using New Outlook and will need to switch to Classic or reach out to IT for support)
Select the account that is being transferred and then click "Change..."
Under settings, drag the bar to the far right and be sure that it says "All" then click Next and Close to exit settings.
In the bottom right-hand corner of the Outlook email home, you will see a progress bar as it downloads emails that were previously only on the server.
Once it is completely downloaded, it should say "All folders are up to date" in the bottom right-hand corner.