Most woodpeckers do not migrate. They are sedentary birds that stay in the same general area all year, but a few species do make seasonal journeys.
Among the true migrants are the yellow-bellied sapsucker, the rufous-bellied woodpecker and the Eurasian wryneck, which breeds in Europe and western Asia and winters in the Sahel region of Africa. In North America, the more northerly populations of the northern flicker, Lewis’s woodpecker and the Williamson’s, red-breasted and red-naped sapsuckers move south in the fall. Most other woodpecker movements are dispersal, such as young birds leaving to find their own territories after fledging, or short eruptions to escape harsh weather. Some species are altitudinal migrants: the grey-capped pygmy woodpecker, for example, moves from hills down to lowlands in winter. The woodpeckers that do migrate travel during the day. So if you see fewer woodpeckers in winter, many have simply stayed put, while sapsuckers and northern flickers are the species most likely to have left.
Source: Wikipedia, “Woodpecker” (Distribution, habitat, and movements)
Image: Yellow-bellied sapsucker in Central Park, by Rhododendrites, CC BY-SA 4.0, via Wikimedia Commons.