In this paper, we develop a framework for an efficient under-approximation of the dynamics of Asynchronous Automata Networks (AANs). An AAN is an Automata Network with synchronised transitions between automata, where each transition changes the local state of exactly one automaton (but any number of synchronizing local states are allowed). The work we propose here is based on static analysis by abstract interpretation, which allows to prove that reaching a state with a given property is possible, without the same computational cost of usual model checkers: the complexity is polynomial with the total number of local states and exponential with the number of local states within a single automaton. Furthermore, we address AANs with classes of priorities, and give an encoding into AANs without priorities, thus extending the application range of our under-approximation. Finally, we illustrate our method for the model checking of large-scale biological networks.