Many species are shifting their ranges in response to global climate change. Range expansions have important genetic and evolutionary consequences, caused by non-random mating at the expanding range margins. One of the consequences is the evolution of dispersal, which increases colonisation speed, provided that a species can adapt sufficiently fast to novel local conditions. Mutation rates can evolve too, under conditions that favor an increased rate of adaptation, but the increase in frequency of a modifier increasing the mutation rate is dependent on genetic hitchhiking on a beneficial mutation at a gene under selection. In sexual populations recombination can then quickly erode the generated linkage disequilibrium. Here we use an individual-based model to show that non-random mating at the expanding range margin practically eliminates the possibility of recombination during invasion, allowing the maintenance of the established linkage disequilibrium. This causes the evolutionary increase of mutation rates, which clearly advances the range expansion both through its effect on the evolution of dispersal rate, and the evolution of local adaptation. This occurs both in a scenario of steadily increasing temperatures across the landscape as with variably increasing temperatures. In contrast, in a spatially stable population, strong directional selection causes the evolution of mutation rates as well as shown in previous theoretical studies, but not when we add variance to the mean selection pressure. By this we show that the evolution of mutation rates possibly plays a more important role in the adaptation of sexual species due to its enhancement by non-random mating during range expansions. This has possibly far-reaching consequences concerning species’ invasiveness and the rate at which species can adapt to novel environmental conditions during range expansions under global climate change.