I am not quite sure how this can happen. When the brush is in the Brushes panel, it stays there until you delete it from the list. Are there any specific brushes that go missing or does it happen to all of them? When this happens again, please send over a screenshot of the whole UI.
For example in screenshot FP4-1 you will notice the Fireball brush does not show at all in the brushes panel, even though I previously imported the brush in. I then grab the brush from your website and drop it into the brushes panel, then these other brushes show up. I now have two copies of the Fireball brush (screenshot FP4-2), however all of the other brushes have disappeared. Can you point me to a video to help navigate your program?
this is not the case of brushes disappearing/re-appering but your Particle system switches between Flame and Follow. The first screenshot you posted is from Flame particle system. When you add the Fireball online brush (which is created in Follow particle system) it will be added into the particular system. Thus the system will change to Follow.
You can see the drop down menu in the Properties panel: