In this paper, we analyze a satellite-DNA family, the RAYSI family, which is specific of the Y chromosomes of Rumex acetosa, a dioecious plant species with a multiple sex-chromosome system in which the females are XX and the males are XY(1)Y(2). Here, we demonstrate that this satellite DNA is common to other relatives of R. acetosa, including Rumex papillaris, Rumex intermedius, Rumex thyrsoides and Rumex tuberosus that are also dioecious species with a multiple system of sex chromosomes. This satellite-DNA family is absent from the genomes of other dioecious Rumex species having an XX/XY sex-chromosome system. Our data confirm recent molecular phylogenies that support a unique origin for all dioecious species of Rumex and two separate lineages for species with single or complex sex-chromosome systems. Our data also support an accelerated degeneration of Y-chromosome in XX/XY(1)Y(2) species by the accumulation of satellite-DNA sequences. On the other hand, the particular non-recombining nature of the Y chromosomes of R. acetosa and their closest relatives lead to a particular mode of evolution of RAYSI sequences. Thus, mechanisms leading to the suppression of recombination between the Y chromosomes reduced the rate of concerted evolution and gave rise to the apparition of different RAYSI subfamilies. Thus, R. acetosa and R. intermedius have two subfamilies (the RAYSI-S and RAYSI-J subfamilies and the INT-A and INT-B subfamilies, respectively), while R. papillaris only has one, the RAYSI-J subfamily. The RAYSI-S and RAYSI-J subfamilies of R. acetosa differ in 83 fixed diagnostic sites and several diagnostic deletions while the INT-A and the INT-B of R. intermedius differ in 27 fixed diagnostic sites. Pairwise comparisons between RAYSI-S and RAYSI-J sequences or between INT-A and INT-B sequences revealed these sites to be shared mutations detectable in repeats of the same variant in same positions. Evolutionary comparisons suggest that the subfamily RAYSI-J has appeared in the common ancestor of R. acetosa and R. papillaris, in which RAYSI-J has replaced totally (R. papillaris) or almost totally the ancestral sequence (R. acetosa). This scenario assumes that RAYSI-S sequences should be considered ancestral sequences and that a secondary event of subfamily subdivision should be occurring in R. intermedius, with their RAYSI subfamilies more closely related to one another than with other RAYSI sequences. Our analysis suggests that the different subfamilies diverged by a gradual and cohesive way probably mediated by sister-chromatid interchanges while their expansion or contraction in number might be explained by alternating cycles of sudden mechanisms of amplification or elimination.