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Recent years have seen an upsurge of interest in dynamical realizations of the superconformal group SU(1, 1|2) in mechanics. Remarking that SU(1, 1|2) is a particular member of a chain of supergroups SU(1, 1|n) parametrized by an integer n, here we begin a systematic study of SU(1, 1|n) multi-particle mechanics. A representation of the superconformal algebra su(1, 1|n) is constructed on the phase space spanned by m copies of the (1, 2n, 2n-1) supermultiplet. We show that the dynamics is governed by two prepotentials V and F, and the Witten-Dijkgraaf-Verlinde-Verlinde equation for F shows up as a consequence of a more general fourth-order equation. All solutions to the latter in terms of root systems reveal decoupled models only. An extension of the dynamical content of the (1, 2n, 2n-1) supermultiplet by angular variables in a way similar to the SU(1, 1|2) case is problematic.