We were told to use 4WD LOW when backing up a hill pushing the RV.... keeps the tranny cooler.
That was AFTER we blew our tranny on the old F250 gasser we had.
The real reason is typically auto trans have fewer clutches in the sections that make the trans reverse. Fewer clutches, maybe less steel and lower operating pressures put more of a load on the trans in reverse when you load the truck.
So, not using 4 wheel low pushing a 12K or more trailer up a hill in reverse may not harm the trans. But using the low will certainly let the truck work a lot easier, and it won't hurt the vehicle.
Now, that said, the trucks with the Ford type axles can really start jumping around when the you start doing some tight turns.
If you don't need the 4WD, only the low range, on the 2011 and newer Fords you can pull the fuse (under the hood) to prevent the auto hubs from locking.
If possible, you should avoid tight turns on hard ground when in either 4WD high or 4WD low. The GM independent suspension is a bit more adept at tight turns in 4WD
I was told, by the Ford salesman when I bought my 2012 F-350, that Chevy's must be at a full stop to shift into 4X4 but Fords can shift on the fly? Not sure if this is correct. I have shifted into 4 high while slowly rolling but have not used 4 low yet. I think I would still be at a complete stop before shifting into 4 low.
The 2011+ Fords are computer controlled. 2WD to 4WD high can be done at any speed, in or out. I typically will jump to 4WD when running into heavy rain. Forcing the front wheels to "drive" helps prevent hydroplaning.
4WD high to low range shifting can be done at low speeds, but the computer will not let the trans shift until safe, usually giving you a dash warning saying stop or shift to neutral.
ATC is turned off in the low range.