Fault stepovers are prime examples of geometric complexity in natural fault zones that may affect seismic hazard by determining whether an earthquake rupture continues propagating or abruptly stops. However, the long‐term pattern of seismicity near‐fault stepovers and underlying mechanisms of rupture jumping in the context of earthquake cycles are rarely studied. Leveraging a hybrid numerical scheme combining the finite element and the spectral boundary integral methods, FEBE, we carry out fully dynamic simulations of sequences of earthquakes and aseismic slip for both compressive and tensile stepovers with off‐fault plasticity. We consider a rate‐and‐state friction law for the fault friction and pressure‐sensitive Drucker–Prager plasticity for the off‐fault bulk response. We observe that the accumulation of plastic deformation, an indication of off‐fault damage, is significantly different in the two cases, with more plastic deformation projected in the overlapping region for the tensile stepover. The seismic pattern for a tensile stepover is more complex than for a compressive stepover, and incorporating plasticity also increases complexity, relative to the elastic case. A tensile stepover with off‐fault plasticity shows rupture segmentation, temporal clustering, and frequent rupture jumping from one fault to another. These results shed light on possible mechanisms of rupture jumping in fault stepovers as well as the long‐term evolution of the fault zone.