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Abstract Coherent structures—flow features that organize material transport and deformation—are central to analyzing complex flows in fluids, plasmas, and active matter. Yet, identifying such structures on dynamic surfaces remains an open challenge, limiting their application to many living and synthetic systems. Here, we introduce a geometric framework to extract Lagrangian and Eulerian coherent structures from velocity data on arbitrarily shaped, time-evolving surfaces. Our method operates directly on triangulated meshes, avoiding global parametrizations while preserving objectivity and robustness to noise. Applying this framework to active nematic vesicles, collectively migrating epithelial spheroids, and beating zebrafish hearts, we uncover hidden transport barriers and Lagrangian deformation patterns—such as dynamic attractors, repellers, isotropic and anisotropic strain—missed by conventional Eulerian analyses. This approach offers a new perspective on soft and living matter, revealing how geometry and activity can be harnessed to program synthetic materials, and how Lagrangian strain and principal deformation directions can help uncover mechanosensitive processes and directional cues in morphogenesis.