If the Priasoft Profile Manager application is run before a mailbox is migrated, or run on a user’s desktop for someone that either will not be migrated or does not have any Outlook profiles pointing to our Exchange server, what happens?

The Priasoft Profile Manger utility is designed to be safely run prior to migration.
When the utility executes, it look at each configured Outlook profile for the current user to see if the details of the profile match with the data stored in the Profile Update File (*.puf).  The PUF contains a list of values that identify mailboxes that have been successfully migrated.
With this pattern, only mailboxes that have been migrated successfully will ever be updated by the Profile Manager.  If a mailbox has not been migrated, failed during migration, or if the PUF is empty or non-existent, the Profile Manager will simply run and then exit, with no work to perform and no changes made to any profiles.
This design was made so that administrators can comfortably and confidently deploy an automation routine, like a logon script, that activates the Profile Manager long before the first mailbox is to be migrated.
In fact, our guidance is to setup such automation, whether a logon script, desktop automation (like SCCM), or other means as soon as possible.  Doing so means that the update mechanism is ready and waiting for the migration to occur and removes a chance of forgetting to enable it at the last minute.  This approach also allows it to be safely run on ALL end-user desktops without complexities of trying to limit it’s use to specific users.
For more detail about the profile manager and deployment scenarios, please refer to the following documents:
Profile Manager User Guide:
https://www.priasoft.com/wp-content/uploads/documents/ProfileManager.pdf

Advanced Profile Manager Guide:
https://www.priasoft.com/wp-content/uploads/documents/Advanced ProfileManager.pdf