Optical methods, which allow the determination of the dominant channels of energy and phase relaxation, are the most universal techniques for the investigation of semiconductor quantum dots. In this paper, we employ the kinetic Pauli equation to develop the first generalized model of the pulse-induced photoluminescence from the lowest-energy eigenstates of a semiconductor quantum dot. Without specifying the shape of the excitation pulse and by assuming that the energy and phase relaxation in the quantum dot may be characterized by a set of phenomenological rates, we derive an expression for the observable photoluminescence cross section, valid for an arbitrary number of the quantum dot's states decaying with the emission of secondary photons. Our treatment allows for thermal transitions occurring with both decrease and increase in energy between all the relevant eigenstates at room or higher temperature. We show that in the general case of N states coupled to each other through a bath, the photoluminescence kinetics from any of them is determined by the sum of N exponential functions, whose exponents are proportional to the respective decay rates. We illustrate the application of the developed model by considering the processes of resonant luminescence and thermalized luminescence from the quantum dot with two radiating eigenstates, and by assuming that the secondary emission is excited with either a Gaussian or exponential pulse. Analytic expressions describing the signals of secondary emission are analyzed, in order to elucidate experimental situations in which the relaxation constants may be reliably extracted from the photoluminescence spectra.