Allisons aren't washed (with cleaning agent) whilst running.... rinsed with just water maybe... but not washed.
This is for all the reasons I've mentioned above.
The second method you've seen is the right one. Washed by motoring over the engine, just as Nubian described.
Yes, it does chuck a load of dirty water out of the exhaust on first start that has pooled in the combustion chamber during the washing procedure.