After the first success in establishing the diffusive, Brownian limit of planar, finite-horizon, periodic Lorentz processes, in 1981 Sinai turned the interest toward studying models when periodicity is hurt, in particular, to locally perturbed Lorentz processes. The 1981 solution for a random-walk-model - by Telcs and the speaker - only led in 2009 to that for the locally perturbed, finite-horizon Lorentz process (by Dolgopyat, Varju and the present author). Beside reporting on these results we also analyze the first steps in extending the limits obtained for the periodic Lorentz process to locally perturbed periodic or quasi-periodic ones (results by Nandori, Paulin, Varju and the speaker).