  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 to 30 per cent of XOC
objects remain bound after anisotropic stellar mass loss imparting a WD natal
kick of $\sim$1 km/s. 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 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.
