  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.
