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ABSTRACT A large fraction of white dwarfs (WDs) have metal-polluted atmospheres, which are produced by accreting material from remnant planetary systems. The composition of the accreted debris broadly resembles that of rocky Solar system objects. Volatile-enriched debris with compositions similar to long-period comets (LPCs) is rarely observed. We attempt to reconcile this dearth of volatiles with the premise that exo-Oort clouds (XOCs) occur around a large fraction of planet-hosting stars. We estimate the comet accretion rate from an XOC analytically, adapting the ‘loss cone’ theory of LPC delivery in the Solar system. We investigate the dynamical evolution of an XOC during late stellar evolution. Using numerical simulations, we show that 1–30 per cent of XOC objects remain bound after anisotropic stellar mass-loss imparting a WD natal kick of $${\sim}1 \, {\rm km \, s^{-1}}$$. We also characterize the surviving comets’ distribution function. Surviving planets orbiting a WD can prevent the accretion of XOC comets by the star. A planet’s ‘dynamical barrier’ is effective at preventing comet accretion if the energy kick imparted by the planet exceeds the comet’s orbital binding energy. By modifying the loss cone theory, we calculate the amount by which a planet reduces the WD’s accretion rate. We suggest that the scarcity of volatile-enriched debris in polluted WDs is caused by an unseen population of 10–$$100 \, \mathrm{au}$$ scale giant planets acting as barriers to incoming LPCs. Finally, we constrain the amount of volatiles delivered to a planet in the habitable zone of an old, cool WD.