This paper provides a framework for argumentation-based persuasion dialogues that enables a participant to implement strategies based on its modelling of its interlocutor's knowledge. The framework is defined on the basis of the recent ASPIC⁺ general model of argumentation, and thus accommodates a range of pos sible instantiations. We extend existing works on persuasion by accounting for both admissible and grounded semantics, and also by allowing participants to not only move arguments that attack those of their interlocutor, but also preferences which undermine the success of these attacks as defeats. We also state formal results for these dialogues, and then use these dialogues to illustrate that appropriate mechanisms for strategising need to account for the logical content of arguments, rather than just rely on their abstract specification.