As an important methodology to measure distribution discrepancy, optimal transport (OT) has been successfully applied to learn generalizable visual models under changing environments. However, there are still limitations, including strict prior assumption and implicit alignment, for current OT modeling in challenging real-world scenarios like partial domain adaptation, where the learned transport plan may be biased and negative transfer is inevitable. Thus, it is necessary to explore a more feasible OT methodology for real-world applications. In this work, we focus on the rigorous OT modeling for conditional distribution matching and label shift correction. A novel masked OT (MOT) methodology on conditional distributions is proposed by defining a mask operation with label information. Further, a relaxed and reweighting formulation is proposed to improve the robustness of OT in extreme scenarios. We prove the theoretical equivalence between conditional OT and MOT, which implies the well-defined MOT serves as a computation-friendly proxy. Extensive experiments validate the effectiveness of theoretical results and proposed model.