Health- I am surprised that you are not familiar with the Hatzalah model. The EMTs usually show up with their personal vehicles with complete BLS equipment. The ambulance is dispatched from a central location, it is likely that the first responders and realized that this patient will benefit from more advanced care so they requested that ALS get dispatched.
The Hatzalah paramedics also respond with their private vehicles with complete ALS gear, this enables them to arrive and initiate care before the ambulance arrives. Is that hard to grasp?