This article concerns testing for differences between groups in many related variables. For example, the focus may be on identifying genomic sites with differential methylation between tumor subtypes. Standard practice in such applications is independent screening using adjustments for multiple testing to maintain false discovery rates. We propose a Bayesian nonparametric testing methodology, which improves performance by borrowing information adaptively across the different variables through the incorporation of shared kernels and a common probability of group differences. The inclusion of shared kernels in a finite mixture, with Dirichlet priors on the different weight vectors, leads to a simple and scalable methodology that can be routinely implemented in high dimensions. We provide some theoretical results, including closed asymptotic forms for the posterior probability of equivalence in two groups and consistency even under model misspecification. The method is shown to compare favorably to frequentist and Bayesian competitors, and is applied to methylation array data from a breast cancer study.