Without meaning to be unresponsive, it depends on what you consider "damage". For example, when you go out and exercise vigorously on a cold morning, you may get some irritation and coughing, and perhaps even have a metallic taste in your throat.  Basically, you've damaged your airways (protective lining has been lost here and there, and the metallic taste is a bit of blood). But not anything beyond a bit of discomfort. Pretty much ANY animal exercising in the cold (horses, dogs, humans) experiences this, and depending on how closely you look, you can find that sort of "damage" at temperatures around freezing. it isn't DAMAGE in the sense that you've permanently injured them and left a scar, but it does count as mild injury to the tissue - the lung is not 100% normal after cold weather exercise.

It gets progressively more severe as the temperatures get colder and/or the exercise becomes harder or longer duration, but if the dog is otherwise normal and healthy, it is rare that the effects get more severe than some mucus on the airways (to protect them from the cold) and a bit of reflex airway narrowing that resolves without treatment. Even at temperatures down below -20F, dogs typically cough a bit but are fine after a night's rest - particularly if they have been routinely exercising and their lungs have somewhat gotten used to the challenge. The serious problems occur when there is already some lung disease and this gets thrown on top of it - or this occurs and then the dog gets exposed to some bug that has an easier time establishing an infection.

Bottom line is that if you are doing a good job of taking care of the dogs and are on top of their health and husbandry needs, no reason to not train and run at temperatures down to about -20F. Past that, it becomes a question of whether you NEED to do it - you can, but there are limits to how much risk is acceptable against whatever benefit is obtained. If it is -35F and you can gain the same benefit 2 days from now when it has gotten a bit warmer.......