For segmented viruses, rapid genomic and phenotypic changes can occur through the process of reassortment, whereby co-infecting strains exchange entire segments creating novel progeny virus genotypes. However, for many viruses with segmented genomes, this process and its effect on transmission dynamics remain poorly understood. Here, we assessed the consequences of reassortment for selection on viral diversity through time using bluetongue virus (BTV), a segmented arbovirus that is the causative agent of a major disease of ruminants. We analysed ninety-two BTV genomes isolated across four decades from India, where BTV diversity, and thus opportunities for reassortment, are among the highest in the world. Our results point to frequent reassortment and segment turnover, some of which appear to be driven by selective sweeps and serial hitchhiking. Particularly, we found evidence for a recent selective sweep affecting segment 5 and its encoded NS1 protein that has allowed a single variant to essentially invade the full range of BTV genomic backgrounds and serotypes currently circulating in India. In contrast, diversifying selection was found to play an important role in maintaining genetic diversity in genes encoding outer surface proteins involved in virus interactions (VP2 and VP5, encoded by segments 2 and 6, respectively). Our results support the role of reassortment in driving rapid phenotypic change in segmented viruses and generate testable hypotheses for in vitro experiments aiming at understanding the specific mechanisms underlying differences in fitness and selection across viral genomes.