Clustered microbiome data have become prevalent in recent years from designs such as longitudinal studies, family studies, and matched case-control studies. The within-cluster dependence compounds the challenge of the microbiome data analysis. Methods that properly accommodate intra-cluster correlation and features of the microbiome data are needed. We develop robust and powerful differential composition tests for clustered microbiome data. The methods do not rely on any distributional assumptions on the microbial compositions, which provides flexibility to model various correlation structures among taxa and among samples within a cluster. By leveraging the adjusted sandwich covariance estimate, the methods properly accommodate sample dependence within a cluster. Different types of confounding variables can be easily adjusted for in the methods. We perform extensive simulation studies under commonly-adopted clustered data designs to evaluate the methods. The usefulness of the proposed methods is further demonstrated with a real dataset from a longitudinal microbiome study on pregnant women.