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Contemporary video stylization approaches struggle to achieve artistic stylization while preserving temporal consistency. While generator-based methods produce visually striking stylized results, they suffer from flickering artifacts in dynamic motion scenarios and require prohibitive computational resources. Conversely, non-generative techniques frequently show either temporal inconsistency or inadequate style preservation.We address these limitations by adapting the physics-inspired transport principles from the Transport-based Neural Style Transfer (TNST) framework (originally developed for volumetric fluid stylization) to enforce inter-frame consistency in video stylization.Our framework employs two complementary transformation fields for artistic stylization: a geometric stylization velocity field governing deformation and an orthogonality-regularized color transfer field managing color adaptations. We further strengthen temporal consistency through two key enhancements to our field architecture: a momentum-preserving strategy mitigating vibration artifacts, and an occlusion-aware temporal lookup strategy addressing motion trailing artifacts. Extensive experiments demonstrate FlowStyler's superior performance across dual dimensions: Compared to generator-based approaches, we achieve 4xlower short-term warping errors, while maintaining comparable style fidelity; Against non-generative methods, FlowStyler attains 22% higher style fidelity with slightly improved temporal stability.