A growing number of young white dwarfs (WDs) with metal-enriched atmospheres are accompanied by excess infrared (IR) emission, indicating that they are encircled by a compact dusty disc of solid debris. Such ‘WD debris discs’ are thought to originate from the tidal disruption of asteroids or other minor bodies. However, the precise mechanism responsible for transporting matter from the disruption radius to the WD surface remains unclear, especially in systems with the highest inferred metal accretion rates g s−1, which cannot be explained by Poynting–Robertson (PR) drag alone. Here we present global time-dependent calculations of the coupled evolution of the gaseous and solid components of WD debris discs. Solids transported inwards (initially due to PR drag) sublimate at tens of WD radii, producing a source of gas that both accretes on to the WD surface and viscously spreads outwards in radius, where it overlaps with the solid disc. Our calculations show that if the aerodynamic coupling between the solids and gaseous discs is sufficiently strong (and/or the gas viscosity sufficiently weak), then gas builds up near the sublimation radius faster than it can viscously spread away. Since the rate of drag-induced solid accretion increases with gas density, this results in a runaway accretion process, as predicted by Rafikov, during which the WD accretion rate reaches values orders of magnitude higher than can be achieved by PR drag alone, consistent with the highest measured values of . We explore the evolution of WD debris discs across a wide range of physical conditions and describe the stages of the runaway process in detail. We also calculate the predicted distribution of observed accretion rates , finding reasonable agreement with the current sample. We use our disc evolution model to show that the steady-state assumption commonly adopted to calculate WD metal accretion rates is inaccurate when the metal settling time in the WD atmosphere is long compared to the viscous time-scale; a long metal settling phase following a runaway accretion event may explain some metal-polluted WDs with no current IR excess. Although the conditions necessary for runaway accretion are at best marginally satisfied given the minimal level of aerodynamic drag between circular gaseous and solid discs, the presence of other stronger forms of solid–gas coupling – such as would result if the gaseous disc is only mildly eccentric – substantially increase the likelihood of runaway accretion.