Local n-particle vertex functions represent the crucial ingredient for diagrammatic extensions of dynamical mean field theory (DMFT). Hitherto their application has been restricted-with a few exceptions-to the n=2-particle vertex while higher-order vertices have been neglected. In this paper we derive a general analytical expression for the local n-particle (one-particle-reducible) vertex of the Falicov-Kimball model (FKM). We observe that the magnitude of such vertex functions itself strongly increases with the number of particles n. On the other hand, their effect on generic Feynman diagrams remains rather moderate due to the damping effect of the Green's functions present in such diagrams. Nevertheless, they yield important contributions to the self-energy corrections calculated in diagrammatic extensions of DMFT as we explicitly demonstrate using the example of dual-fermion calculations for the two-dimensional FKM at quarter filling of the stationary f electrons. Here corrections to the self-energy obtained from the three-particle vertex are indeed comparable in magnitude to corresponding corrections stemming from the two-particle vertex.