<p>Peridynamics (PD) theory has shown great promise in fracture simulation due to its ability to accommodate discontinuities in the displacement field. Due to its simplicity and computational efficiency, explicit time-stepping is a popular way for simulating unsteady-state peridynamic (PD) problems, particularly those involving extensive material failure. Unlike conventional partial differential equation problems, the time step size in PD models is only restricted by the horizon size rather than the spatial grid size, owing to the special form of its Courant-Friedrichs-Lewy (CFL) condition. For dual-horizon PD models, where the horizon varies to deal with multiscale phenomenon with coarse and fine resolved regions, global time-stepping schemes could be very efficient when material points possess highly different horizon sizes. In this paper, we propose a family of high-order explicit local time-stepping (LTS) schemes for dual-horizon bond-based and ordinary stated-based PD based on the Runge-Kutta method. This approach allows for fine time steps in regions of interest to enhance accuracy, while the remainder of the domain can be solved with coarser time steps. We provide a rigorous convergence analysis for the proposed LTS schemes and perform numerical experiments to verify the expected convergence orders. Additionally, the PD simulation results, including those of plates with pre-existing cracks and the 3D Kalthoff-Winkler experiment, demonstrate effectiveness of the proposed method in achieving accurate fracture simulations.</p>