Leakage currents through high-permittivity thin films are one of the major bottlenecks impeding the downscaling of integrated circuits. Thus, a fundamental, physical understanding of their origin is highly desirable. In the present work microscopic physical models for relevant leakage current mechanisms are developed and embedded in a kinetic Monte Carlo simulation framework that allows qualitative and quantitative analyses of the electronic transport. Three-dimensional transport simulations are carried out at the level of single charge carriers. Spatially localized phenomena, such as percolation of charge carriers across point-like defects, being subject to structural relaxation processes, or electrode roughness effects, are investigated in this simulation scheme. A self-consistent, closed transport model for the technologically very important TiN/ZrO₂ material system is developed and viable system optimization strategies are derived.