Feature attributions often hide a critical modeling choice: they explain a prediction along a counterfactual path from a reference state to an input. Different baselines, interpolations, and generative trajectories define different paths and can therefore produce different explanations. We study this path ambiguity as a modeling problem. Our central question is whether the path can be chosen by a distribution-level transport process, rather than by a hand-designed interpolation or by the sensitivity geometry of the model being explained. We separate attribution into fixed-path credit allocation and path selection. For a fixed path, we prove that the Aumann-Shapley line integral is the unique attribution rule under fixed-path axioms and coordinate-trace separability. For path selection, we minimize kinetic action over flows that transport a reference distribution to the data distribution, yielding a transport-geodesic attribution principle. We instantiate this framework using Rectified Flow and derive stability bounds quantifying how vector-field approximation errors affect the resulting attributions. Experiments show that Reflow improves cross-seed attribution stability, while learned transport paths yield spatially coherent explanations with competitive deletion faithfulness among the evaluated path-based methods. Our code is available at https://github.com/cenweizhang/OTFlowSHAP.

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