The theory of neoclassical transport in an impure toroidal plasma is extended to allow for larger pressure and temperature gradients and faster toroidal rotation than are usually considered. Under these conditions, the density of heavy impurities is not constant on flux surfaces, and the neoclassical transport becomes a nonlinear function of the gradients. Rapid toroidal rotation increases the transport, which can significantly exceed the conventional Pfirsch–Schluter value if the impurity Mach number is of order unity. In a plasma with steep density or temperature profile, the transport is severely reduced and can even be a nonmonotonic function of the gradients. Finally, in the presence of both rapid toroidal rotation and steep gradients, the transport becomes sensitive to the geometry of the magnetic equilibrium. For instance, in a single-null diverted magnetic field the ion particle flux is typically inward if the ion drift is toward the X -point and changes direction if the toroidal field is reversed.