We present and investigate a general framework for studying modal fixpoint logics and some related versions of monadic second-order logic, by means of certain finite automata that operate on Kripke structures. Characteristic of these modal automata is that the co-domain of their transition function is a set of formulas of a so-called one-step logic. The motivation for taking this perspective is that if a logic is characterised by a class of modal automata, many of its properties are already determined at the level of the much simpler one-step logic.