Deterministic particle approximations are one of the possible ways to construct solutions to scalar conservation laws, to nonlinear diffusion equations, and to some Wasserstein gradient flows. Despite their finite-dimensional nature, they can catch certain ”smoothig-effect” properties of the relevant PDE. I will review some recent results on this aspect, focusing on discrete counterparts of two celebrated properties: the Oleinik one-sided Lipschitz property for scalar conservation laws and the Aronson-Bénilan estimate for the porous medium equation. We will also consider the case of the one-dimensional nonlocal interaction equation with repulsive Morse potential. The results involve M. Schmidtchen (TU Dresden), V. Iorio (L’Aquila), M. D. Rosini (Chieti-Pescara), and D. Matthes (TU Munich) as co-authors. This seminar is part of the activities of the Excellence Department Project CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.