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 indirect excitons. Considering both the normal-phase and condensate regimes of the exciton subsystem, we investigate the transmobility 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. The temperature dependence of the transmobility and its dependence on the interlayer separation are studied. It is shown 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 transmobility 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.