Each week it's a good idea to search for families using Family Explorer or Classic View for New Register families. There is no mechanism for parishes to be notified of families who have been added to your church nor moved to another church.
When you add a family, pull a family over to your parish, or use the on line link for families to register themselves, you should go into Gabriel and review and update their status.
Options to find and review this information.
Family Explorer Search
Filter the Church ID column by your Church ID, and Filter the Status column by "new"
Classic View Search
Click Search from below the blue menu, choose New Register in the Status Field and Current parish.
The Report - Family Status with Registration Status