A geosynchronous spaceborne-airborne bistatic synthetic aperture radar (GEO SA-BSAR) system is an important technique to achieve long-time moving target monitoring over a wide area. However, due to special bistatic configuration of GEO SA-BSAR, two major challenges, i.e., severe range migration and space-variant Doppler parameters for moving targets, hinder the moving target indication (MTI) processing. Traditional SAR MTI methods, which do not take the challenges into consideration, will defocus the moving targets, leading to a loss of the signal-to-noise ratio (SNR). To focus moving targets and estimate motion parameters accurately, long-time coherent integration space-time adaptive processing (LTCI-STAP) is proposed for GEO SA-BSAR MTI in this paper. First, a modified adaptive spatial filtering based on the bistatic signal model is performed to suppress the clutter. Then, an LTCI filter bank is constructed to achieve range migration correction and moving target focusing, which yields the optimal output signal and filtering parameters. Finally, constant false alarm rate (CFAR) detection is carried out to determine the targets, and the space-variant Doppler parameters, solved from the filtering parameters, are used for estimating moving target positions and velocities. Simulations verify the effectiveness of our method.