We add extreme driving to the Kob-Andersen kinetically constrained lattice-gas model in order to mimic the effect of gravity on dense granular systems. For low particle densities, the current that develops in the system agrees at arbitrary field intensity with a mean-field theory. At intermediate densities, spatial correlations give rise to nonmonotonic dependence of the current on field intensity. At higher densities, the current ultimately vanishes at a finite, field-dependent density. We supplement the study of this bulk behavior with an investigation of the current through a narrow hole. There, lateral flow decreases the local density in front of the hole. Remarkably, the current through the hole quantitatively agrees with a theoretical prediction based on the bulk current at the measured local density.