Rapid species turnover in tropical mountains has fascinated biologists for centuries. A popular explanation for this heightened beta diversity is that climatic stability at low latitudes promotes the evolution of narrow thermal tolerance ranges, leading to local adaptation, evolutionary divergence and parapatric speciation along elevational gradients. However, an emerging consensus from research spanning phylogenetics, biogeography and behavioural ecology is that this process rarely, if ever, occurs. Instead, closely related species typically occupy a similar elevational niche, while species with divergent elevational niches tend to be more distantly related. These results suggest populations have responded to past environmental change not by adapting and diverging in place, but instead by shifting their distributions to tightly track climate over time. We argue that tropical species are likely to respond similarly to ongoing and future climate warming, an inference supported by evidence from recent range shifts. In the absence of widespread in situ adaptation to new climate regimes by tropical taxa, conservation planning should prioritize protecting large swaths of habitat to facilitate movement.