ABSTRACTA quarter of DA white dwarfs are metal polluted, yet elements heavier than helium sink down through the stellar atmosphere on time-scales of days. Hence, these white dwarfs must be the currently accreting material containing heavy elements. Here we consider whether the scattering of comets or asteroids from an outer planetary system, following stellar mass-loss on the asymptotic giant branch, can reproduce these observations. We use N-body simulations to investigate the effects of stellar mass-loss on a simple system consisting of a planetesimal belt whose inner edge is truncated by a planet. Our simulations find that, starting with a planetesimal belt population fitted to the observed main-sequence evolution, sufficient mass is scattered into the inner planetary system to explain the inferred heavy element accretion rates. This assumes that a fraction of the mass scattered into the inner planetary system ends up on star-grazing orbits, is tidally disrupted and is accreted on to the white dwarf. The simulations also reproduce the observed decrease in accretion rate with cooling age and predict accretion rates in old (>1 Gyr) white dwarfs, in line with observations. The efficiency we assumed for material scattered into the inner planetary system to end up on star-grazing orbits is based on a solar-like planetary system, since the simulations show that a single planet is not sufficient. Although the correct level of accretion is reproduced, the simulations predict a higher fraction of accreting white dwarfs than observed. This could indicate that the evolved planetary systems are less efficient in scattering bodies on to star-grazing orbits or that dynamical instabilities post-stellar mass-loss cause rapid planetesimal belt depletion for a significant fraction of systems.