Not all organisms that feel the climate change around them will move. Some will make the best of their new situation and stay put. These organisms won't be very mobile, like oak trees. They'll also be genetically diverse, with some individuals in the population adapted to slightly warmer than the local temperature, and some adapted to slightly cooler. Evolution will then have variation to work on. What will happen next is a shift in success. "If the area warms, warmer-adapted organisms that were struggling under cooler temperatures will survive better, have more kids and become favored."
Life & Earth Science, "10 Predictions for Evolution Over the Next Million Years", [online], Available: http://science.discovery.com/life-earth-science/10-predictions-for-evolution.htm