Summary: We present a general triangular array central limit theorem for \(U\)-statistics, where the kernel \(h_k (x_1, \dots, x_k)\) and its dimension \(k\) may increase with the sample size. Motivating examples that require such a general result are presented, including a class of Hodges-Lehmann estimators, subsampling estimators, and combining \(p\)-values using data splitting. A result for the so-called \(M\)-statistic is also presented, which is defined as the median of some kernel computed over all subsets of the data of a given size. The conditions in the theorems are verified in the motivating examples as well.