Sex-determining systems often undergo high rates of turnover but for reasons that remain largely obscure. Two recent evolutionary models assign key roles, respectively, to sex-antagonistic (SA) mutations occurring on autosomes and to deleterious mutations accumulating on sex chromosomes. These two models capture essential but distinct key features of sex-chromosome evolution; accordingly, they make different predictions and present distinct limitations. Here we show that a combination of features from the two models has the potential to generate endless cycles of sex-chromosome transitions: SA alleles accruing on a chromosome after it has been co-opted for sex induce an arrest of recombination; the ensuing accumulation of deleterious mutations will soon make a new transition ineluctable. The dynamics generated by these interactions share several important features with empirical data, namely, (i) that patterns of heterogamety tend to be conserved during transitions and (ii) that autosomes are not recruited randomly, with some chromosome pairs more likely than others to be co-opted for sex.