It still works if they pick up other dropped lmgs. Commandos will fire the ober lmg34 on the move, normal troops will not.
the fire on the move is an attribute tied to the weapon.
Here's how it works:
The pick up item (the item you see on the ground), literally morph into different weapon depending on who pick it up, but this system isn't bug free. Similar case with the weapon rack
The commandos bren dropped on the ground and picked up by the ober is the same bren gun used by tommies. The ober can't fire the bren on the move.
This is how it actually work mechanically behind the scene. The observed effect is similar but it doesn't work the way you think.
Firing on the move is not an attribute you can just turn on and off for the weapon.