The recent advances of new genomic technologies has enabled to identify and characterize sex chromosomes in an increasing number of non-model species, revealing that many plants and animals undergo frequent sex chromosome turnovers. What evolutionary forces drive these turnovers remains poorly understood, but it was recently proposed that drift might play a more important role than generally assumed. We analyzed the dynamics of different types of turnovers using individual-based simulations, and show that when mediated by genetic drift, turnovers are usually easier to achieve than substitutions at neutral markers, but that their dynamics and relative likelihoods vary with the type of the resident and emergent sex chromosome system (XY and/or ZW), and the dominance relationships among the sex-determining factors. Focusing on turnovers driven by epistatically dominant mutations, we find that drift-mediated turnovers that preserve the heterogamety pattern are 2-4x more likely than those along which the heterogametic sex changes. This ratio nevertheless decreases along with effective population size, and can even reverse in case of extreme polygyny. This can be attributed to a “drift-induced” selective force, known to influence transitions between male and female heterogamety, but which according to our study, does not affect turnovers that preserve the heterogametic sex.