A re-evaluation of time-averaged accretion rates at DBZ-type white dwarfs points to historical, time-averaged rates significantly higher than the currently observed episodes at their DAZ counterparts. The difference between the ongoing, instantaneous accretion rates witnessed at DAZ white dwarfs, which often exceed 108 g s−1, and those inferred over the past 105–106 yr for the DBZ stars can be of a few orders of magnitude, and therefore must result from high-rate episodes of tens to hundreds of years so that they remain undetected to date. This paper explores the likelihood that such brief, intense accretion episodes of gas-phase material can account for existing data. For reasonable assumptions about the circumstellar gas, accretion rates approaching or exceeding 1015 g s−1 are possible, similar to rates observed in quiescent cataclysmic variables, and potentially detectable with future X-ray missions or wide-field monitoring facilities. Gaseous debris that is prone to such rapid accretion may be abundant immediately following a tidal disruption event via collisions and sublimation, or if additional bodies impinge upon an extant disc. Particulate disc matter accretes at or near the Poynting–Robertson drag rate for long periods between gas-producing events, consistent with rates inferred for dusty DAZ white dwarfs. In this picture, warm DAZ stars without infrared excesses have rates consistent with accretion from particulate discs that remain undetected. This overall picture has implications for quasi-steady state models of accretion and the derived chemical composition of asteroidal debris in DBZ white dwarfs.