Car with auto gearbox.When cruising down hill,you slip the gear shift to 'N".will this action damage the AUTO

Wont hurt the tranny , but it WILL cause your brakes to wear out sooner. Leaving it in gear will help keep your speed lower, and shifting into N will not save enough gas to measure.

it will take many thousands of miles of downshifting to cause major harm AS LONG AS HE DOES NOT DO IT TOO SOON. he is at more risk of blowing the engine than hurtin the trans. downshift too soon and the car will slow but engine will speed up. trannys are very tough. i don't think I've ever had a car or truck that i didn't downshift and they were all autos.

Newer trannys should be fine. I do this in my 88 cadillac & it has a stock 4 speed auto tranny. i've herd some older trannys wont re-engage into gear if you're moving at a high rate of speed. but newer cars that have tiptronic "manual-matic" vavle bodies should handle the task with no problem. I do notice my car goes into third then 4th then engages the torque converter lockup. so i only do it on verly long hills.

no it shouldn't hurt anything but what is the point? its not saving you any fuel as soon as the wheel speed exceeds the engine speed the tranny goes into free spin anyway. what i'm saying is in a standard tranny staying in gear makes engine revv up on long down hill right? but not with automatic right? car goes back down to idle until you get to bottom of hill. so why bother taking the chance of damaging your tranny?

putting it in neutral wont damage it, but dropping it back into gear while moving will. autos arent designed for that, and dropping into neutral is supposed to only be used in an emergency technically, but if you just come to a complete stop it wont hurt. if you put it in drive from neutral itll eventually start slipping gears.

No, so long as you leave the engine running. The engine operates the transmissions oil pump.

If you switch the engine off the transmission will no longer have it's own pressure and damage will definitely occur.