The characterization of the out-of-equilibrium dynamics of quantum many body systems is one of the most interesting challenges of modern quantum statistical physics: one of the main issues is to investigate quantities
and techniques which can encode huge amount of informations into a transparent form, giving insight into the physical properties of the system. For free systems evolving after a quantum quench, the quasiparticle picture has proven to be very effective at providing the evolution of several entropy-related quantities at the ballistic scale. In this talk I will discuss some recent work on a novel operatorial approach to the study of the quasiparticle picture, which allows to obtain effective hydrodynamic scale expressions for the dynamics of the entanglement hamiltonians and negativity hamiltonians, and also to easily extend the quasiparticle picture to novel configurations and quantities of interest, such as systems involving projective measurements. Finally, I will discuss tentative extensions to interacting integrable systems.