Let us modify the scatterer configuration of a planar, finite-horizon Lorentz process in a bounded domain. Sinai asked in 1981 whether, for the diffusively scaled variant of the modified process, convergence to Brownian motion still holds. The main result of this work answers Sinai's question in the affirmative. Other types of local perturbations are also investigated: finite-horizon periodic Lorentz processes in the half strip or in the half plane (in these models, the local perturbation is the boundary condition) and finite-horizon, periodic Lorentz processes with a small, compactly supported external field in the strip. The corresponding limiting processes are Brownian motions with suitable boundary conditions and the skew Brownian motion on the line. The proofs combine Stroock and Varadhan's martingale method in [SV1] with our recent work in [DSV]