  We study kinetic Monte-Carlo (KMC) descriptions of active particles. By
relying on large discrete time steps, KMC algorithms accelerate the
relaxational dynamics of active systems towards their steady-state. We show,
however, that their continuous-time limit is ill-defined, leading to the
vanishing of trademark behaviors of active matter such as the motility-induced
phase separation, ratchet effects, as well as to a diverging mechanical
pressure. We show how mixing passive steps with active ones regularizes this
behavior, leading to a well-defined continuous-time limit. We propose new AKMC
algorithms whose continuous-time limits lead to the active dynamics of
Active-Ornstein Uhlenbeck, Active Brownian, and Run-and-Tumbles particles.
