The problem isn't either of those if you are EOCing for a reasonable distance(under 20 or so miles at a time).
The only issue I have with EOC in an automatic is that you don't have any hydraulic pressure to get the transmission back into gear and you have to manually restart the engine. I'd love to be able to bump start the engine at the end of an EOC on my automatic.
In my car, as long as the engine stays above 400 or so rpm (yes, 400) it can hold a gear with the engine off but once it falls below that the transmission lets go.
I think the main harm done to automatics from EOC is the fact that the insides of the transmission are spinning from the driveshaft turning, but since the pump isn't operating there's little or no lubrication.
Most vehicles have a maximum distance they can be flat towed before the engine needs to be started and idled in neutral for a bit to relube the transmission. You should always go by this on the EOC speed and time.
The tracker manual specifies every 150 miles or 2 hours. I don't think that EOCing for a couple of miles is going to hurt my car much.
On the other hand, Saturn specs alone vary widely. Their 4 banger FWD transmissions say not to tow more than 1 mile any faster than 5mph if the front wheels are on the ground while some of their 5 speed auto 4wd vehicles can be towed at 65 mph or one hour and at the extreme end they have a few models that can be towed for 7 hours straight regardless of speed.
It's my understanding that, in common automatic transmissions, they still have a high enough oil level when off to stay lubricated but without circulating that oil the heat builds up and fries the oil/transmission.
It's my understanding that, in common automatic transmissions, they still have a high enough oil level when off to stay lubricated but without [b]circulating[b] that oil the heat builds up and fries the oil/transmission.