We study optical pulse propagation through a linear, dispersive, gain-loss-assisted bulk medium whose refractive index is time-varying. To analyze the dynamics, we have used a novel technique of time transformation that provides universal formulas of pulse propagation. Our analytical and numerical investigations reveal that optical pulses show asymmetric behavior while propagating in opposite direction through such a medium, in both the temporal and spectral domains. Moreover, the wavelength shift during this process is the most interesting outcome which is limited in range, but could be tuned by varying the refractive index with time. Phenomena that are observed in this Letter are novel and realizable in practical devices such as coupled waveguides where the refractive index is a function of time. © 2019 Optical Society of America