Im not sure what car you have now BUT you dont want to shift in to neutral to save gas. It uses zero gas when coasting if you leave it in gear, on newer cars.

 

yah, the injectors shutdown when engine braking so best to keep it in gear...

not sure about our AT's but does it feed enough oil on the gearbox
when in neutral yet coasting? some cars you are told not to put it in
neutral cause it deprives the gears from oil. and hot gears is no good.

 

Its called DFCO (deceleration fuel cut off) and yes, my current vehicle does it to, but on just slight downgrades, DFCO would slow me down to much, making me need to step on the gas again, while Neutral coasting lets me coast all the way.

I've never heard of Neutral coasting problems with the engine running, only with the engine off. I dont know for sure though.