The paper analyses multi-agent strategic dialogues on possibly infinite argumentation frameworks. We develop a formal model for representing such dialogues , and introduce FO A-ATL, a first-order extension of alternating-time logic, for expressing the interplay of strategic and argumentation-theoretic properties. This setting is investigated with respect to the model checking problem, by means of a suitable notion of bisimulation. This notion of bisimu-lation is also used to shed light on how static properties of argumentation frameworks influence their dynamic behaviour.