To perform the Unmanned Aerial Vehicle-based (UAV-based) maritime object tracking task, we deliberately acquire a new ship tracking data set, namely, UAVSeaShip102. This task faces severe challenges due to rapid target motion, drastic scale variations, and frequent occlusion in complex aerial scenes. Recent transformer-based trackers have shown remarkable potential in modeling long-range dependencies. However, their performance often deteriorates when faced with large- scale changes or unreliable large-scale updates. To address these issues, we propose a Dynamic Multi-scale Fusion Transformer Tracker (DMFTrack) for UAV-based single object tracking, which comprises two key components, i.e., a Dynamically-Weighted Multi-scale Feature Fusion Module (DWMS-FFM) and an Adaptive Template Update Mechanism (ATUM). The DWMS- FFM adaptively aggregates features from multiple scales via a learnable gating strategy, allowing the model to dynamically balance spatial details and global contextual information. Furthermore, the ATUM employs a Score Head to estimate the reliability of the current template through dual-stage cross-attention, allowing confidence-guided online template updates that mitigate model drift. Our DMFTrack achieves consistent improvements in both accuracy and robustness. Extensive experiments conducted on UAVSeaShip102 and six commonly-used UAV-based benchmarks demonstrate that our method outperforms 23 trackers in challenging scenarios, which involve scale variation, occlusion, and background clutter, while maintaining real-time performance.