A significative fraction of high-mass stars sail away through the interstellar medium of the galaxies. Once they evolved and died via a core-collapse supernova, a magnetised, rotating neutron star (a pulsar) is usually left over. The immediate surroundings of the pulsar is the pulsar wind, which forms a nebula whose morphology is shaped by the supernova ejecta and channelled into the circumstellar medium of the progenitor star in the pre-supernova time. Irregular pulsar-wind nebulae display a large variety of radio appearances, screened by their interacting supernova blast wave, or harbour asymmetric up–down emission. Here, we present a series of 2.5-dimensional (2 dimensions for the scalar quantities plus a toroidal component for the vectors) black non-relativistic magneto-hydrodynamical simulations exploring the evolution of the pulsar-wind nebulae generated by a red supergiant and a Wolf-Rayet massive supernova progenitor, moving with Mach number $M=1$ and $M=2$ into the warm phase of the Galactic plane. black In such a simplified approach, the progenitor's direction of motion, the local ambient medium magnetic field, and the progenitor and pulsar axis of rotation, are all aligned; this restricted our study to peculiar pulsar-wind nebula of high-equatorial-energy flux. We find that the reverberation of the termination shock of the pulsar-wind nebulae, when sufficiently embedded into its dead stellar surroundings and interacting with the supernova ejecta, is asymmetric and differs greatly as a function of the past circumstellar evolution of its progenitor, which reflects into their projected radio synchrotron emission. This mechanism is particularly at work in the context of remnants involving slowly moving or very massive stars. We find that the mixing of material in plerionic core-collapse supernova remnants is strongly affected by the asymmetric reverberation in their pulsar-wind nebulae.