We study the high-dimensional properties of an invading front in a disordered medium with random pinning forces. We concentrate on interfaces described by bounded slope models belonging to the quenched Kardar-Parisi-Zhang [Phys. Rev. Lett. 56, 889 (1986)] universality class. We find a number of qualitative transitions in the behavior of the invasion process as dimensionality increases. In low dimensions [Formula Presented] the system is characterized by two different roughness exponents: the roughness of individual avalanches and the overall interface roughness. We use the similarity of the dynamics of an avalanche with the dynamics of invasion percolation to show that above [Formula Presented] avalanches become flat and the invasion is well described as an annealed process with correlated noise. In fact, for [Formula Presented] the overall roughness is the same as the annealed roughness. In very large dimensions, strong fluctuations begin to dominate the size distribution of avalanches; this phenomenon is studied on the Cayley tree, which serves as an infinite dimensional limit. We present numerical simulations in which we measured the values of the critical exponents of the depinning transition, both in finite-dimensional lattices with [Formula Presented] and on the Cayley tree, which support our qualitative predictions. We find that the critical exponents in [Formula Presented] are very close to their values on the Cayley tree and we conjecture on this basis the existence of a further dimension, where mean-field behavior is obtained.