We study the time-dependent spectra produced via the bulk Compton process by a cold, relativistic shell of plasma moving (and accelerating) along the jet of a blazar, scattering on external photons emitted by the accretion disc and reprocessed in the broad-line region (BLR). The bulk Comptonization of disc photons is shown to yield a spectral component contributing in the far-ultraviolet band, and would then be currently unobservable. On the contrary, the bulk Comptonization of broad-line photons may yield a significant feature in the soft X-ray band. Such a feature is time-dependent and transient, and dominates over the non-thermal continuum only when: (i) the dissipation occurs close to, but within, the BLR; and (ii) other competing proce...