Current language-guided robotic manipulation systems often require low-level action-labeled datasets for imitation learning. While object-centric flow prediction methods mitigate this issue, they remain limited to scenarios involving rigid objects with clear displacement and minimal occlusion. In this work, we present Embodiment-Centric Flow (EC-Flow), a framework that directly learns manipulation from action-unlabeled videos by predicting embodiment-centric flow. Our key insight is that incorporating the embodiment's inherent kinematics significantly enhances generalization to versatile manipulation scenarios, including deformable object handling, occlusions, and non-object-displacement tasks. To connect the EC-Flow with language instructions and object interactions, we further introduce a goal-alignment module by jointly optimizing movement consistency and goal-image prediction. Moreover, translating EC-Flow to executable robot actions only requires a standard robot URDF (Unified Robot Description Format) file to specify kinematic constraints across joints, which makes it easy to use in practice. We validate EC-Flow on both simulation (Meta-World) and real-world tasks, demonstrating its state-of-the-art performance in occluded object handling (62% improvement), deformable object manipulation (45% improvement), and non-object-displacement tasks (80% improvement) than prior state-of-the-art object-centric flow methods. For more information, see our project website at https://ec-flow1.github.io .