A danger in moving cold hands or other body parts is that the elasticity of the muscles and tendons are compromised because they are in a contracted state and if you try to move cold body parts fast, you can cause damage to the tissue such as micro tears and pulls to either the muscles or the tendons. Slow movement and in the medium range of motion is always advised when the extremities are cold.
When the body is cold, the blood is kept near the core vital organs and circulation is slowed to the extremities such as the hands and feet. That makes our hands and fingers feel cold and stiff. Stretching is not a solution and our teachers and coaches have been teaching this mistruth about stretching for years.
When you stretch beyond the mid range of motion you are creating micro tears in the muscle tissue or tendons and the body's response is to rush blood to that site to both immobilize and repair the damage. This process gives us the sensation of "warming up" when in reality we are damaging our tissue structures. Whenever we move, we must only move as far as the mid range of motion, not the extreme where we will stretch, tear and damage tissue.
There are actually two categories of muscle, fast twitch and slow twitch. Musicians should take the time to learn which ones are which and how to utilize them in their craft. Even so, forcing fast twitch muscles to move fast or to stretch them when they are cold and in their contracted state could damage them. Think of your muscles as being like warm gravy. The gravy can easily pour out of the bowl when it is warm. Now put the bowl in the fridge for half an hour and note that it no longer pours fluidly. You can't just run a spoon through the gravy to warm it up, it needs to come out of the fridge and be warmed totally.
If you have a teacher or coach who prescribes stretching and isolation exercises to warm up the body, find another teacher. It is not their fault that they have been given erroneous information themselves from their own teachers, but, their ignorance on the subject can cause you permanent damage. Ignorance is not bliss if it results in tendon, nerve or muscular disorders. That is like going to a mechanic who says that your tires are bald but you can probably get away on them for another few months. He may be the best mechanic in the world but he is risking your life.
If one wishes to truly warm up the body and consequently the hands, one needs to sit in a warm room so that the whole body warms up, not just the part they are going to use. Another solution is to do some mild whole body movement to get the blood pumping throughout the circulatory system.
There are mini steppers on the market for under $50 that a musician can take to a gig with them and use in the green room before a performance. After doing twenty minutes or about 2,000 steps on one of those, the blood will be circulating efficiently throughout the whole body and one may even break a small sweat. You won't have to warm up your legs because you've already been walking all day and, movement that is well known such as walking is as simple as the brain turning on and off a switch. One doesn't need to warm up to remember how to ride a bike. The brain just knows what to do, like flicking a switch.
Conditioning is important, too. If you can only do three minutes on the stair stepper before fatigue sets in then you're not going to achieve a full body warm up in that amount of time so, it would behoove you to do this every day so the body is conditioned to work at that level without fatigue. One doesn't want to go on stage exhausted and weak. It is also advised to be hydrated before, during and after this simple body warm up procedure.
I'll not endorse any particular brand but you can find mini steppers on Ebay, tax and shipping free. Read the user reviews on Amazon to find a brand you think you can trust.