Abstract We study the relic abundance of several stable particles from a generic dark sector, including the possible presence of dark asymmetries. After discussing the different possibilities for stabilising multi-component dark matter, we analyse the final relic abundance of the symmetric and asymmetric dark matter components, paying special attention to the role of the unavoidable conversions between dark matter states. We find an exponential dependence of the asymmetries of the heavier components on annihilations and conversions. We conclude that having similar symmetric and asymmetric components is a natural outcome in many scenarios of multi-component dark matter. This has novel phenomenological implications, which we briefly discuss.