We propose a kinetic theory to describe the power dependence of the photocurrent lineshape in optically pumped quantum dots at low temperatures in both zero and finite magnetic fields. We show that there is a crossover power P-c, determined by the electron and hole tunneling rates, where the photocurrent no longer reflects the exciton lifetime. For P > P-c, we show that the photocurrent saturates due to the slow hole escape rate, whereas the linewidth increases with power. We analyze the spin-doublet lineshape in high magnetic fields and determine to what measure it reflects the degree of circular polarization of incident light.