Investigation of the kinetic principle of a diode-pumped alkali vapor laser (DPAL) is key to achieve multifunctional DPALs. In this work, we propose a spatiotemporal model, which combines the time-dependent rate equations of population densities and propagation equations of energies to study the dynamic processes from turn-on to steady-state in DPALs. Time evolution of population densities and pump and laser intensity are resolved on a picosecond time scale to study the build-up process of laser oscillations. For nanosecond-pulse pumping, we obtain a laser pulse of 1.6 ns and a delay time of 2.6 ns at an incident pulse width of 2 ns. This pulse can be stretched by increasing the pump pulse width and delayed by applying a more extended cavity. For CW operation, spiking and relaxation oscillations resulting from a dynamic balance of gain and losses are demonstrated to be much faster than other types of lasers.