As an engineer that deals professionally with material issues and how they can impact equipment functionality, the stock answer is that you operate your gear within the specifications provided by the manufacturer.
Common sense goes a long way to answering this question though. Very low temps do indeed push the limits on what most manufacturers will 'qualify' their equipment to operate safely in. Keep in mind there are other environmental factors besides temperature that can be part of that equation (in my world, the radiation field), but the two main issues are indeed the operating temperature and relative humidity of the local environment. A less prevalent issue, but probably no less important to this particular question, is how long the gear is exposed to the extreme environment. Sometimes gear will be qualified to operate in the extreme environment, but only for a limited amount of time. (Over time, for example, corrosion due to the environment may cause the equipment not to operate properly, but in the initial stages corrosion might not be an issue. Another factor which is time dependent is fatigue...more on fatigue later.)
Applying some common sense to the question, in a cold weather environment I'd be more concerned with moving gear from inside to outside than the other way around...with respect to moisture intrusion, going from liquid to solid is a much larger concern that the reverse. In an extreme hot environment, I'd be concerned with moving gear from outside to inside, but less so than in a cold environment, since there's no potential for a moisture phase change, which can be damaging due to both the expansion and contraction of the moisture from solid to liquid and vice versa, but also the damage that can be caused when solid ice forms inside of gear in the case of accumulated moisure or standing water. Ice accumulation can cause potential for displacement of interrnal components, or perhaps in a worst case scenario to electronics the cracking of circuit boards).
With respect to circuit boards, exterme temperature changes can cause expansion and contraction. If there's no space margin provided to allow for that, bowing of the board can result or again, in a worst case scenario board delamination or cracking. Or course, it could be argued that the inside of your gear will stay warm due to ohmic heating, but my discussion is worst case.
However, with all of that said, I think my main concern in exteme cold would indeed be the mic diaphragms...especailly mylar diaphragms. Even though there's probably not much movement in a mic diaphragm, these things are razor thin, so it wouldn't take much to damage them. The concern I'd have is called high cycle fatigue (think about working a coat hanger to the point that it breaks...each back and forth movement is one cycle). Probably safe to assume diaphragms don't have a high cycle fatigue concern at normal temps. However, what about once you start lowering the temp below manufacturer specs? I really can't say for sure if it happens, but it's definitely conceiveable. If diaphragms do fatigue, over long time periods you could get some degree of diaphragm failure (longer times equals higher fatigue cycles...thus the term high cycle fatigue). I'd think this would be particularly true of mylar diaphragms, which intuitively I'd say become progressively more brittle at lower and lower temperatures. It's probably not as big a concern for metal diaphragms.
The bottom line is...I'm pretty sure that you won't find many manufacturers of electronic equipment that would qualify or warrant the performance of electronic gear for long term functionality below 32F / 0C specifically because of the concern for internal/trapped moisture going solid upon freezing, unless it's specifically designed for those temps.
Personally if faced with the decision, I'd assess my diaphragms and keep mylar out of the extreme cold. Beyond that, I'd probably use my gear outside for short time periods based on the comments people have made here that they haven't experienced issues, but say if I was going to snoe.down and was going to be outside for three straight days, if the weather forecast called for really cold temps, I'd probably consider leaving my best stuff inside.