In this paper, we consider the nonparametric Behrens-Fisher problem with cluster correlated data. A class of weighted Mann-Whitney test statistics are introduced and studied. In particular, a comparison with other recent testing procedures for related problems is provided. The new tests are valid when the distributions do not have the same scales and/or shapes under the null hypothesis. A general class of weighted U-statistics for clustered data, encompassing the Mann-Whitney statistic, is also introduced. A simulation studies the type I error robustness and the power of the new and of some recently proposed procedures. This study shows that the incorporation of appropriate weights can greatly improve the power of the test. A real data example illustrate the use of the tests.