We study Lorentz processes in two different settings. Both cases are characterized by infinite expectation of the free-flight times, contrary to what happens in the classical Gallavotti-Spohn models. Under a suitable Boltzmann-Grad type scaling limit, they converge to non-Markovian random-flight processes. A further scaling limit yields another non-Markovian process, i.e., a superdiffusion obtained by a suitable time-change of Brownian motion. Furthermore, we obtain the governing equations for our random flights and anomalous diffusion, which represent a non-local counterpart for the linear-Boltzmann and diffusion equations. It turns out that these have the form of fractional kinetic equations in both time and space. To prove such results, we develop a technique based on averaging Feller semigroups.