The detection of ground-moving objects in aerial videos has evolved over the years to handle more challenges such as large camera motion, the small size of the objects, and occlusion. Recently, aerial detection has been attempted using principal component pursuit (PCP) due to its superiority in detecting small moving objects. However, PCP-based detection methods generally suffer from high-false detections as well as high-computational loads. This paper presents a novel PCP-based detection method called kinematic regularization with local null space pursuit (KRLNSP) that drastically reduces false detections and the computational loads. KRLNSP models the background in an aerial video as a subspace that spans a low-dimension subspace while it models the moving objects as moving sparse. Accordingly, the detection is achieved by using multiple local null spaces and enhanced kinematic regularization. The multiple local null spaces allow real-time execution to nullify the background while preserving the moving objects unchanged. The kinematic regularization penalizes these moving objects to filter out false detections. The extensive evaluation of KRLNSP and relevant current state-of-the-art methods prove that the KRLNSP outperforms these methods (the true positive rate of KRLNSP is 98% and its false positive rate is 0.4%) and significantly reduces the computational loads (KRLNSP execution time is 0.3 s/frame).