Because the elements are much harsher around you durning winter, your skin is more likely to dry out. This can lead to itchy skin if you don't moisturize it properly. Itchy skin can then turn into eczema.
Always dress your children in cotton clothing that is next to the skin. Cotton fabrics allow your skin to breath and this will avoid skin chaffing from the friction that wool fabrics can cause. You will also want to consider changing your detergent that the clothing is washed in for the winter months. Depending on the type of laundry detergent you use, this could add to your dry skin problem.
Just because it is winter does not mean you can forget about sunscreen. Sunscreen will protect them from the sun's UV rays and also give an extra protective barrier against the environmental elements that can affect the skin.