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Most likely they use the fact that the shot location moved, not the engine noise.

How? With multiple shots it's easy as hell. But from one it's quite hard?

You could hear the firing pin hitting the cartridge (harder to hear than engine running) then you should hear the gasses exiting barrel somewhere around 0,001 second later. Then calculate doppler... Sounds kinda hard.

I highly doubt they're providing that service for single shots. Besides, I can't see too many thugs firing a single shot from a car in commission of an assault or homicide. They're not exactly known for their accuracy at the best of times.

I bet the doppler shift from two different listening microphones would give you a pretty easily calculation for how fast the gun was moving (at least relative towards/away speeds with respect to the two microphones) - I'd guess 3 or more microphones listening to the same gunshot could pinpoint both direction and speed.

I was thinking about the Doppler too, but the Doppler shift of the bullet while in flight doesn't really do it. And the time-frame to do it while inside gun is really really small.

Ah, that makes sense.

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