We derive a kinetic theory capable of dealing both with large spin-orbit coupling and Kondo screening in dilute magnetic alloys. We obtain the collision integral nonperturbatively and uncover a contribution proportional to the momentum derivative of the impurity scattering S matrix. The latter yields an important correction to the spin diffusion and spin-charge conversion coefficients, and fully captures the so-called side-jump process without resorting to the Born approximation (which fails for resonant scattering), or to otherwise heuristic derivations. We apply our kinetic theory to a quantum impurity model with strong spin-orbit, which captures the most important features of Kondo-screened Cerium impurities in alloys such as Ce_{x}La_{1-x}Cu_{6}. We find (1) a large zero-temperature spin-Hall conductivity that depends solely on the Fermi wave number and (2) a transverse spin diffusion mechanism that modifies the standard Fick's diffusion law. Our predictions can be readily verified by standard spin-transport measurements in metal alloys with Kondo impurities.