Deheuvels (1981a,b,c) and Genest and Rémillard (2004) have shown that powerful rank tests of multivariate independence can be based on combinations of asymptotically independent Cramér–von Mises statistics derived from a Möbius decomposition of the empirical copula process. A result on the large-sample behavior of this process under contiguous sequences of alternatives is used here to give a representation for the limiting distribution of such test statistics and to compute their relative local asymptotic efficiency. Local power curves and asymptotic relative efficiencies are compared under familiar classes of copula alternatives.