  The metal budgets in some white dwarf (WD) atmospheres reveal that
volatile-rich circumstellar bodies must both exist in extrasolar systems and
survive the giant branch phases of stellar evolution. The resulting behaviour
of these active asteroids or comets which orbit WDs is not well-understood, but
may be be strongly influenced by sublimation due to stellar radiation. Here we
develop a model, generally applicable to any extrasolar system with a main
sequence or WD star, that traces sublimation-induced orbital element changes in
approximately km-sized extrasolar minor planets and comets traveling within
hundreds of au. We derive evolution equations on orbital timescales and for
arbitrarily steep power-law sublimation dependencies on distance, and place our
model in a Solar system context. We also demonstrate the importance of coupling
sublimation and general relativity, and the orbital consequences of outgassing
in arbitrary directions. We prove that nongravitational accelerations alone
cannot result in orbit crossing with the WD disruption radius, but may shrink
or expand the orbit by up to several au after a single pericentre passage,
potentially affecting subsequent interactions with remnant debris and planets.
Our analysis suggests that extant planets must exist in polluted WD systems.
