For general matched-pair data with polytomous responses in biomedical research, the Stuart–Maxwell test and the Bhapkar (1966) test are commonly used for evaluating marginal homogeneity. For data collected in clusters, we propose extensions for statistical inference without structural within-cluster correlation or distributional assumptions. Meanwhile, two extended Obuchowski tests are proposed based on the work of Obuchowski (1998) generally applied to clustered matched-pair binary data. A Monte Carlo simulation study illustrates that our proposed extension to the Stuart–Maxwell test and the two extended Obuchowski tests perform well with respect to the power and the nominal size, though the extended Bhapkar test is asymptotically equivalent to the other three tests, it is not recommended in practice due to its being liberal in the nominal size.