Climate change is redistributing marine and terrestrial species globally. Life‐history traits mediate the ability of species to cope with novel environmental conditions, and can be used to gauge the potential redistribution of taxa facing the challenges of a changing climate. However, it is unclear whether the same traits are important across different stages of range shifts (arrival, population increase, persistence). To test which life‐history traits most mediate the process of range extension, we used a 16‐year dataset of 35 range‐extending coral‐reef fish species and quantified the importance of various traits on the arrival time (earliness) and degree of persistence (prevalence and patchiness) at higher latitudes. We show that traits predisposing species to shift their range more rapidly (large body size, broad latitudinal range, long dispersal duration) did not drive the early stages of redistribution. Instead, we found that as diet breadth increased, the initial arrival and establishment (prevalence and patchiness) of climate migrant species in temperate locations occurred earlier. While the initial incursion of range‐shifting species depends on traits associated with dispersal potential, subsequent establishment hinges more on a species’ ability to exploit novel food resources locally. These results highlight that generalist species that can best adapt to novel food sources might be most successful in a future ocean.