We investigate the behavior of discrete systems on a one-dimensional lattice composed of localized units interacting with each other through nonlocal, nonlinear reactive dynamics. In the presence of second-order and third-order steps coupling two or three neighboring sites, respectively, we observe, for appropriate initial conditions, the propagation of waves which subsist in the absence of mass transfer by diffusion. For the case of the third-order (bistable) model, a counterintuitive effect is also observed, whereby the homogeneously less stable state invades the more stable one under certain conditions. In the limit of a continuous space the dynamics of these networks is described by a generic evolution equation, from which some analytical predictions can be extracted. The relevance of this mode of information transmission in spatially extended systems of interest in physical chemistry and biology is discussed.