I will discuss the emergence of effective evolution equations for the dynamics of N interacting quantum particles, of fermionic type, starting from the many-body Schroedinger equation. In the last years, there has been a lot of progress in the derivation of the Hartree-Fock (HF) equation, a nonlinear evolution equation for weakly correlated initial states, in the mean-field regime. There, one typically considers initially confined particles, with density that grows linearly with N. In this talk, I will discuss the case of extended systems, at fixed particle density. The HF dynamics arises after extracting the mean-field behavior at the local scale, which is made possible thanks to the propagation of a suitable local semiclassical structure of the initial datum along the HF flow. In the second part of the talk I will adapt the strategy to the case of many-body systems where the number of particles is not fixed, in the mean-field regime. I will present the rigorous derivation of the Hartree-Fock-Bogoliubov equation, a nonlinear equation for the dynamics of initial data characterized by a nonzero pairing density, relevant for the description of superconductors in BCS theory. Based on joint works with L. Fresta and B. Schlein, and with S. Marcantoni and J. Sabin.