Its not what it looks like. Airports are entirely different animals. Basically you have to have special permission and training just to drive on the field, and that's to work there. The PAPD has their own set of sogs. When its determined that FDNY is needed, the units are staging at their main PAFD building. (JFKs 269. Im sure LGA has a similar staging point) They are led out to the incident as the PAPD or airport ops units can talk directly to the tower for permissions. And go straight to the scene. The 51( JFK 1st due bat) and probably the 1st due to LGA I beleive has a tower or PAPD radio in their truck, so they can coordinate on the way or while staging. What you dont want, is 20 some odd fd units driving around trying to find the best way to an incident, while at the same time aircraft may still be moving trying to get back to gates, or being directed away from the incident. Even sitting still, a jet engine idling can ruin your day. Even more so if you hit an aircraft.
Both LGA and JFK have training drills every so often. Kennedys is Operation Safe and I forget what LGAs is called but its full sized mock mci plane crash with "victims" made up to look real. I've done several with the scouts at both. They run the whole thing like real over 2 or 3 hours. PA comes out, puts out some real fire, FD is escorted onto the field and puts out the rest, and all of the victims are triaged etc. They get along very well. It just looks different from the outside.