Migratory instincts developed in different species for different reasons, but for the most part they are responses to population pressure. Most migration follows the "leave somewhere cold for somewhere warm, then come back in the summer" pattern. So why would a species live somewhere that got too cold for them in the first place?
The first camp suggests that animals first lived where it was warm all year long, and they didn't need to migrate. As the population grew, resources became scarce. During warm months, northern latitudes were relatively hospitable, so some members of the species expanded their range and began living in these areas. When winter came, food grew scarce and it got too cold for them, so they temporarily relocated to warmed latitudes [source: Drickamer].
The second camp says that climate change is responsible. Species that lived in the north were able to live there all year during periods when the climate was warm enough. As tens of thousands of years passed, however, the climate gradually changed, and eventually the winters grew too cold, forcing the species to head south each year.
The truth about migration is probably a mix of the two, and it probably differs by species. However, the first theory is most likely -- population pressure is the driving force behind most migrations, and, in fact, most evolution. Climate change may have lent a hand in forming or shaping migratory patterns, but it was not the primary force.