  We develop a microscopic theory of the Coulomb drag effect in a hybrid system
consisting of spatially separated two-dimensional quantum gases of degenerate
electrons and dipolar excitons. We consider both the normal-phase and
condensate regimes of the exciton subsystem and investigate the cross-mobility
of the system being the kinetic coefficient, which couples the static electric
field applied to the electron layer with the particle density current (flux) in
the exciton subsystem. We study the temperature dependence of the
cross-mobility and its dependence on the interlayer separation. We show that
exciton-exciton interaction plays a dramatic role. If the exciton gas is in the
normal phase, then the screening of interlayer interaction by the exciton
subsystem results in an exponential damping of the cross-mobility with the
decrease of temperature, while at low temperatures, the interactions result in
a robust bosonic transport due to the emergence of the Bogoliubov
quasiparticles.
