Kinetic transport equations with a given confining potential and non-linear relaxation type collision operators are considered. General (monotone) energy dependent equilibrium distributions are allowed with a chemical potential ensuring mass conservation. Existence and uniqueness of solutions is proven for initial data bounded by equilibrium distributions. The diffusive macroscopic limit is carried out using compensated compactness theory. The result are drift-diffusion equations with nonlinear diffusion. The most notable examples are of porous medium or fast diffusion type, with exponent ranging from 0 to 5/3, in dimension 3.