Species need to move to keep pace with changing climates, but we do not know if species can move at the required speed. Spread rates of native species may underestimate how fast species can move, we therefore assessed how fast Lessepsian species (marine non-native species that invaded the Mediterranean from the Red Sea through the Suez Canal) can spread to give a ‘best-case’ assessment of the effects of climate change on marine biodiversity. We show that about 20% of Lessepsian species could not spread fast enough to keep pace with climate change in about 20% of the global seas and this suggests that climate change may lead to biodiversity loss. The velocity of climate change on the seabed is much lower than at the sea surface, and as a result of this the proportion of species that keep pace with climate change at the seabed was much larger than at the sea surface. This suggests that locations at depth could act as refuges for slow dispersing species. Our analysis compared different estimates of the spreading abilities of marine species and showed that the estimate of spread rates strongly affects the predicted effect of climate change on biodiversity. Providing more accurate estimates of the spreading ability of marine species should therefore have priority if we want to predict the effect of climate change on marine biodiversity. This study is a first approximation of the potential scale and distribution of global marine biodiversity loss and can provide benchmark estimates of the spread rates that species could achieve in colonizing suitable habitat. Assisted colonization may be required to maintain biodiversity in the most strongly affected areas.