Species can either adapt to new conditions induced by climate change or shift their range in an attempt to track optimal environmental conditions. During current range shifts, species are simultaneously confronted with a second major anthropogenic disturbance, landscape fragmentation. Using individual-based models with a shifting climate window, we examine the effect of different rates of climate change on the evolution of dispersal distances through changes in the genetically determined dispersal kernel. Our results demonstrate that the rate of climate change is positively correlated to the evolved dispersal distances although too fast climate change causes the population to crash. When faced with realistic rates of climate change, greater dispersal distances evolve than those required for the population to keep track of the climate, thereby maximizing population size. Importantly, the greater dispersal distances that evolve when climate change is more rapid, induce evolutionary rescue by facilitating the population in crossing large gaps in the landscape. This could ensure population persistence in case of range shifting in fragmented landscapes. Furthermore, we highlight problems in using invasion speed as a proxy for potential range shifting abilities under climate change.