A complete, kinetic description of electron-seeded strong-field QED showers in crossed electromagnetic fields is derived. The kinetic structure of the shower and its temporal evolution are shown to be a function of two parameters: the initial shower quantum parameter and radiation time. The latter determines the short and long time evolution of the shower. Explicit solutions for the shower multiplicity (number of pairs per seed electron) and the emitted photon spectrum are obtained for both timescales. Our approach is first derived considering showers in a constant, homogeneous magnetic field. We find that our results are valid for any crossed fields and we apply them to laboratory settings for which we obtain fully analytical, predictive scaling laws.