We have studied the influence of a square array of pinning centers on the dynamics of vortex avalanches in Pb thin films by means of ac- and dc-magnetization measurements. Close to the superconducting transition T-c, the commensurability between the vortex lattice and the pinning array leads to the well known local increments of the critical current. As temperature T decreases, matching features progressively fade out and eventually disappear. Further down in temperature, vortex avalanches develop and dominate the magnetic response. These avalanches manifest themselves as jumps in the dc magnetization and produce a lower ac shielding, giving rise to a paramagnetic reentrance in the ac screening chi(')(T). Within the flux-jump regime, two subregimes can be identified. Close to the boundary where vortex avalanches develop, the field separation between consecutive jumps follows the periodicity of the pinning array and a field- and temperature-dependent screening is observed. In this regime, the response also depends on frequency f in agreement with theoretical models for magnetothermal instabilities. At low enough temperatures and fields, the screening saturates to a constant value independent of T, H, and f, where jumps are randomly distributed. We have also found that vortex instabilities occupy a larger portion of the H-T diagram in patterned samples than in films without nanoengineered pinning sites. Finally, we discuss the possible origin of the vortex avalanches and compare our results with previous experimental and theoretical studies.