We consider the randomization of correlated, translationally invariant distributions of indistinguishable particles on lattices by random hopping, possibly involving several jump distances with generally different rates (and where double occupancy is excluded). Probabilities for various subconfigurations of n empty sites satisfy infinite closed sets of linear equations (for each n) in which the generator of the dynamics is self-adjoint. We provide a detailed spectral analysis of this generator for the two-point probabilities (or corresponding correlations) on a one-dimensional lattice. For just nearest-neighbor (1NN) and second-nearest-neighbor (2NN) jumps, the dynamics changes smoothly as a function of the ratio of the 2NN- to 1NN-jump rates up to the critical value ¼, where there is a nonanalytic transition in the dynamical structure. Generalizations are indicated.