We study the dynamics of countable groups on their respective spaces of quasimorphisms. For cohomologically non-trivial quasimorphisms we show that there are no invariant measures and classify stationary measures. Within the equivalence class of any given quasimorphism we find both uniquely stationary orbit closures which are in fact boundaries and orbit closures with uncountably many ergodic stationary probability measures. We apply these results to study hulls of uniform approximate lattices which arise from twists by quasimorphisms. We show that these hulls do not admit invariant probability measures (extending results by Machado and Hrushovski) and classify stationary probability measures on these hulls.