We present a new approach to the description of the output from a frequency-shifted feedback (FSF) laser seeded by a phase-fluctuating but stationary continuous-wave (CW) laser. We illustrate the new analysis by showing how short frequency-chirped pulses arise for appropriate operating conditions. We show the equivalence of two common viewpoints of the FSF laser output as either a moving comb of equidistant frequencies or as a fixed set of discrete frequencies. We also consider operation of a FSF laser when there is no external seeding laser, and instead the cavity radiation originates with spontaneous emission.