Connected and irreducible components in families

For an algebraic stack x flat and of finite presentation over a scheme S, we introduce various notions of relative connected components and relative irreducible components. The main distinction between these notions is whether we require the total space of a relative component to be open or closed in x. We study the representability of the associated functors of relative components, and give an application to the moduli stack of curves of genus g admitting an action of a fixed finite group G.