Complicated spectrotemporal processes are associated with the generation of signal and idler output pulses on a nanosecond timescale in an injection-seeded optical parametric oscillator (OPO). The mechanisms of such spectrotemporal dynamics are revealed by numerical simulation, including innovative modeling of instantaneous-frequency profiles and frequency chirp. These simulations are in satisfactory agreement with optical-heterodyne measurements of output from a nanosecond-pulsed OPO system that is based on periodically poled ${\mathrm{KTiOPO}}_4$ , pumped at $532\mathrm{nm}$ by a $\mathrm{Nd}:\mathrm{YAG}$ laser and injection-seeded at a signal wavelength of $\sim 842\mathrm{nm}$ . Frequency chirp in narrowband signal output pulses from such an OPO system has previously been observed to depend on phase mismatch between the pump, signal, and idler waves, and also on the pump-pulse energy. Our simulations accurately predict this behavior and yield realistic estimates of the frequency chirp, optical bandwidth, and spectral purity of the signal output pulse as it evolves, including effects that are not readily observed directly. This approach provides insight into instrumental conditions that facilitate continuously tunable, single-longitudinal-mode operation of such a pulsed OPO system, with optical bandwidth as close as possible to the Fourier-transform limit.