  Inspired by the recent Kepler discoveries of circumbinary planets orbiting
nine close binary stars, we explore the fate of the former as the latter evolve
off the main sequence. We combine binary star evolution models with dynamical
simulations to study the orbital evolution of these planets as their hosts
undergo common-envelope stages, losing in the process a tremendous amount of
mass on dynamical timescales. Five of the systems experience at least one
Roche-lobe overflow and common-envelope stages (Kepler-1647 experiences three),
and the binary stars either shrink to very short orbits or coalesce; two
systems trigger a double-degenerate supernova explosion. Kepler's circumbinary
planets predominantly remain gravitationally bound at the end of the
common-envelope phase, migrate to larger orbits, and may gain significant
eccentricity; their orbital expansion can be more than an order of magnitude
and can occur over the course of a single planetary orbit. The orbits these
planets can reach are qualitatively consistent with those of the currently
known post-common-envelope, eclipse-time variations circumbinary candidates.
Our results also show that circumbinary planets can experience both modes of
orbital expansion (adiabatic and non-adiabatic) if their host binaries undergo
more than one common-envelope stage; multiplanet circumbinary systems like
Kepler-47 can experience both modes during the same common-envelope stage.
Additionally, unlike Mercury orbiting the Sun, a circumbinary planet with the
same semi-major axis can survive the common envelope evolution of a close
binary star with a total mass of 1 MSun.
