[en] We study numerically and theoretically the d-dimensional Hamiltonian motion of fast particles through a periodic field of scatterers, modelled by bounded, localized (time-dependent) potentials that we refer to as (in)elastic non-dissipative Lorentz gases. We illustrate the wide applicability of a random walk picture previously developed for a field of scatterers with random spatial and/or time dependence by applying it to four other models. First, for a periodic array of spherical scatterers in d ≥ 2, with a smooth (quasi-)periodic time dependence, we show Fermi acceleration: the ensemble averaged kinetic energy (||p(t)||²) grows as t^2/5. Nevertheless, the mean squared displacement (||q(t)||²) ∼ t² behaves ballistically. These are the same growth exponents as for random time-dependent scatterers. Second, we show that in the soft elastic and periodic Lorentz gas, where the particles' energy is conserved, the motion is diffusive, as in the standard hard Lorentz gas, but with a diffusion constant that grows as ||p₀||⁵, rather than only as ||p₀||. Third, we note the above models can also be viewed as pulsed rotors: the latter are therefore unstable in dimension d ≥ 2. Fourth, we consider kicked rotors, and prove them, for sufficiently strong kicks, to be unstable in all dimensions with (||p(t)||²) ∼ t and (||q(t)||²) ∼ t³. Finally, we analyse the singular case d = 1, where (||p(t)||²) remains bounded in time for time-dependent non-random potentials, whereas it grows at the same rate as above in the random case.