How often does the credit bureau update beacon scores and by how many points?

Credit bureaus don't update your credit scores. They update your credit reports. It is a third party company (FICO, Beacon) who processes the information on your credit reports and produces your credit scores.

I have continuous real-time access to my Equifax score, and it updated once at the end of February, and then increased by a point in Mid-March. So I'm going to say twice a month based on my experience with accessing my Equifax Fico Score. I can't be certain about the other two bureaus.

Basic credit information is reported and updated (usually once a month, sometimes less frequently) by the creditors. Your credit score is a calculation based on that data. The calculation is only performed when a score is requested. So, your score would be updated only upon request.