Satellite videos provide continuous observations of surface dynamics but pose significant challenges for multi-object tracking (MOT), especially under unstabilized conditions where platform jitter and the weak appearance of tiny objects jointly degrade tracking performance. To address this problem, we propose DeTracker, a joint-detection-and-tracking framework tailored for unstabilized satellite videos. DeTracker introduces a task-driven Global-Local Motion Decoupling (GLMD) module to address the motion imbalance between dominant platform motion and weak target motion. It suppresses background-dominated motion via global semantic alignment at the feature level and captures target-specific motion through local refinement, improving trajectory stability and identity consistency. In addition, a Temporal Dependency Feature Pyramid (TDFP) module is developed to perform cross-frame temporal feature fusion, enhancing the continuity and discriminability of tiny-object representations. We further construct a new benchmark dataset, SDM-Car-SU, which simulates multi-directional and multi-speed platform motions to enable systematic evaluation of tracking robustness under varying motion perturbations. Extensive experiments on both simulated and real unstabilized satellite videos demonstrate that DeTracker significantly outperforms existing methods, achieving 61.1% MOTA on SDM-Car-SU and 45.3% MOTA on real satellite video data. The code and dataset will be publicly available at https://github.com/alex-chenjiajun/DeTracker.