The goal of multi-object tracking is to detect and track all objects in a scene while maintaining unique identifiers for each, by associating their bounding boxes across video frames. This association relies on matching motion and appearance patterns of detected objects. This task is especially hard in case of scenarios involving dynamic and non-linear motion patterns. In this paper, we introduce DeepMoveSORT, a novel, carefully engineered multi-object tracker designed specifically for such scenarios. In addition to standard methods of appearance-based association, we improve motion-based association by employing deep learnable filters (instead of the most commonly used Kalman filter) and a rich set of newly proposed heuristics. Our improvements to motion-based association methods are severalfold. First, we propose a new transformer-based filter architecture, TransFilter, which uses an object's motion history for both motion prediction and noise filtering. We further enhance the filter's performance by careful handling of its motion history and accounting for camera motion. Second, we propose a set of heuristics that exploit cues from the position, shape, and confidence of detected bounding boxes to improve association performance. Our experimental evaluation demonstrates that DeepMoveSORT outperforms existing trackers in scenarios featuring non-linear motion, surpassing state-of-the-art results on three such datasets. We also perform a thorough ablation study to evaluate the contributions of different tracker components which we proposed. Based on our study, we conclude that using a learnable filter instead of the Kalman filter, along with appearance-based association is key to achieving strong general tracking performance.