Generalizing manipulation policies across robot embodiments remains difficult because standard policies tie the interactions needed for a task to the actions of a particular robot. We study zero-shot cross-embodiment manipulation, where a policy trained on source embodiments must be deployed on a structurally different target embodiment without additional task demonstrations. We introduce Kinematic Interaction Transfer across Embodiments (KITE), which decouples manipulation into embodiment-agnostic interaction prediction and embodiment-specific kinematic decoding, connected through a learned representation of interaction intent based on contact patterns. An interaction policy is trained to predict the intent using demonstration data only collected on a source robot, and shared across morphologically different target robots. A kinematic decoder realizes these intents by learning from synthetic data generated from each embodiment's kinematic model. Through such decoupling, KITE can robustly transfer learned skills to a new embodiment given only its kinematic model, without recollecting any task-specific demonstration data. We evaluate KITE on four manipulation tasks in simulation and the real world spanning transfer between parallel grippers, dexterous hands, and composite embodiments. KITE successfully achieves zero-shot transfer to structurally different target embodiments, and consistently outperforms state-of-the-art baselines. Project website: https://kite-manip.github.io/.

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