Average-T-matrix and coherent medium theories are used to study the motion of localized excitations on substitutionally disordered lattices. We derive equations which relate coherent medium results for bond and site averaging and show how these reduce to the two-body solution results of Gochanour, Andersen, and Fayer. Numerical results for Po(t), the probability of remaining at the origin for two-dimensional nearest-neig~abor lattices are presented.