A study of the propagation of localized waves in collisionless plasma media is undertaken. It is shown how such pulsed wavefields can be generated from finite-time dynamic apertures. We demonstrate that localized waves exhibit an unusual robustness when they propagate in a plasma medium modeled by the Klein-Gordon equation. The decay rates of the centroids of such pulses are comparable to those of localized waves traveling in vacuum. Furthermore, when a parameter controlling the spatiotemporal coupling of the spectral components of the localized wave is tuned to the plasma frequency, the decay of the centroid of the radiated field becomes slower than that of a localized wave traveling in vacuum. We provide an explanation of this behavior based on the unusual depletion of the spectral components of the finite-time localized waves.