Climate change stressors (e.g., warming and ocean acidification) are an imminent challenge to the physiological performance of marine organisms. Several studies spanning the last decade have reported widespread effects of warming and acidification on marine fishes, especially teleosts, but more work is needed to elucidate the responses in marine elasmobranchs, i.e., sharks and rays. Dispersal capacity, as a result of locomotor performance, is a crucial trait that will determine which group of elasmobranchs will be more or less vulnerable to changes in the environment. In fact, efficient and high locomotor performance may determine the capacity for elasmobranchs to relocate to a more favorable area. In this review we integrate findings from work on locomotion of marine sharks and rays to identify characteristics that outline potential vulnerabilities and strength of sharks and rays under climate change. Traits such as intraspecific variability in response to climatic stressors, wide geographic range, thermotaxis, fast swimming or low energetic costs of locomotion are likely to enhance the capacity to disperse. Future studies may focus on understanding the interacting effect of climatic stressors on morphology, biomechanics and energetics of steady and unsteady swimming, across ontogeny and species.