In the presence of persistent payoff heterogeneity, the evolution of the aggregate strategy hugely depends on the underlying strategy composition under many evolutionary dynamics, while the aggregate dynamic under the standard BRD reduces to a homogenized smooth BRD, where persistent payoff heterogeneity averages to homogeneous transitory payoff shocks. In this paper, we consider deterministic evolutionary dynamics in heterogeneous population and develop the stronger concept of local stability by imposing robustness to persistent payoff heterogeneity. It is known that nonaggregability holds generically if the switching rate in a given evolutionary dynamic correlates with the payoff gain from a switch. To parameterize the payoff sensitivity of an evolutionary dynamic, we propose to use tempered best response dynamics with bounded support of switching costs.