This paper investigates dynamics that persist under isotopy in classes of orientation-preserving homeomorphisms of orientable surfaces. The persistence of periodic points with respect to periodic and strong Nielsen equivalence is studied. The existence of a dynamically minimal representative with respect to these relations is proved and necessary and sufficient conditions for the isotopy stability of an equivalence class are given. It is also shown that most the dynamics of the minimal representative persist under isotopy in the sense that any isotopic map has an invariant set that is semiconjugate to it.