Not all birds migrate to warm countries for wintering. Some birds also migrate form really cold areas to spent winter in the Caucasus.
Or they just migrate to lower altitudes. And many bird species do not change the area at all. They are called resident birds.
How birds protect against icy wind and snow
Birds have many superb adaptations that allow them to survive even in the most frigid conditions. Generally, birds are warm-blooded animals that have a much higher metabolism, and thus higher body temperature, than humans. But, they have also many physical and behavioral adaptations to keep warm. Some bird species grow extra feathers as part of a late fall molt to give them thicker protection in winter. Like their migratory friends, they fuel up in autumn for extra energy to generate body heat. The dense plumage keeps birds warm. If it gets icy birds fluff out their feathers, which makes them look like a feather ball. This creates air pockets for additional insulation. Therefor downs are a very popular filling material for anoraks, sleeping bags and duvets! To save energy, birds reduce their amount of movements. Many small birds gather in large flocks at night and crowd together in a small, tight space to share body heat. Maybe in shrubbery and trees, but nesting boxes are often taken up as well. And this is only a selection of their strategies. When temperatures start to dip, it isn’t necessary to worry about how birds keep warm; they have plenty of efficient adaptations to survive even the chilliest nights. However, if it gets really icy, or you live in an area where snow will cover the grounds, there are ways how you can support them.
How to support birds in winter
If you want to provide food, select seeds, suet, nuts, items high in fat and calories to give birds plenty of energy to generate more body heat. Provide shelter! Evergreen shrubs and coniferous trees provide suitable shelter throughout the winter. A brush pile can add instant shelter without needing to plant. Adding brush piles to your property is also a way to practice effective habitat conservation by reusing materials rather than just burning or sending them to a landfill. Brush piles give birds a safe, sheltered place to roost. Here in Georgia most birds still find suitable nesting places. Therefor, do not be disappointed, if the nesting box you might have placed has not been taken up in spring. It will be highly welcomed as shelter during cold nights.