You could get condensation that will short out your equipment. General practice is to turn it off before moving and wait for it to warm up to room temperature.
Condensation is the problem. What I do is seal the cold gear in zip lock (or other types) plastic bags, while in the cold environment. Then bring them into the warm environment and let them warm up to room temperature before taking them out of the plastic. My understanding why it is important to seal them in plastic, or other air tight container, is that the colder air sealed in the bags is at a lower relative humidity, and as it warms up, no moisture can enter causing condensation on the gear, which will warm up slower than the air in the bag. I swear by this technique and have shot in very cold environments with video cameras and have not had any issues with the gear after it warms up.