Carrier transport through electrically active grain boundaries is studied under high-electric-field conditions. Electrons trapped at the interface and screened by ionized shallow and deep bulk defects are responsible for the formation of double Schottky barriers which reduce the carrier flow by several orders of magnitude. At large applied bias, electric fields up to 1 MV/cm can build up near the interface, leading to the generation of hot electrons and to the subsequent production of holes by impact ionization. This process is studied in a realistic model, taking the inhomogeneous field as well as longitudinal-optic and -acoustic phonon scattering into account. The hot-electron distribution function is calculated as the solution of a Fokker-Planck equation in energy space. With a reasonable estimate for the pair-creation rate we determine the yield for hole production near the interface. These minority carriers are swept back to the grain boundary where they serve as an additional screening charge for the electrons. We determine the steady-state and dynamic behavior of the barrier, including the holes in our calculations. The accumulation of holes at the interface can lead to the breakdown of the barrier which, in its most dramatic form, can even result in a bistability. The dynamic behavior of the barrier reflects the presence of holes through the development of a negative capacitance, in agreement with experimental observations. The negative capacitance is explained by the finite recombination time of the holes at the interface. In a model where the recombination is substituted by hole emission, a realistic current-voltage characteristic but no negative capacitance is found.