The 'Visible Migration' of birds that we see at certain times of year doesn't just involve birds leaving our country, but also birds of the same species passing through 'on passage' from one country to another, and birds staying in the UK but re-locating from one area to another.
Meadow Pipits that we see at this time of year can originate from areas north of the UK, and are merely utilising our land mass as a 'stepping-stone'. Also, Meadow Pipits that breed on mountain and moorland will seek out lower altitudes during the harsh weather of Winter. This can be why we see larger numbers of them on salt-marshes and coastal areas during Winter.
Many other birds migrate, not just the obvious ones. Large numbers of Robins and Chaffinches will migrate from the Continent to the UK each Winter, as some of ours move south. The Robin you see in the garden in Winter isn't necessarily the same one that was there in Summer!
Here's an extract to help explain:
Meadow Pipits
Anthus pratensis migrate south in autumn through Britain. The origins and destinations of these birds are unclear but they may well comprise a mixture of populations from Iceland, the Faeroes, Norway and northern Britain, with many of them moving on to winter in France, Iberia and northern Africa (Tom Dougall, ‘Meadow Pipit’, The Migration Atlas: movements of the birds of Britain and Ireland, (Wernham, C V, Toms, M P, Marchant J H, Clark J A, Siriwardena G M and Baillie S R, eds) T & A D Poyser, London (2002) pp.470-473).