<p>This paper concerns with monotone systems on spaces of vector-valued functions, which are defined on group bundles. Under the assumption of asymptotic smoothness, we develop theory on their asymptotic propagation. By refining geometrical features of the propagation region and extending the dynamical method of Weinberger, we obtain the asymptotic annihilation, asymptotic persistence, and upward convergence for discrete-time monotone systems with pointwise asymptotic smoothness. Then we provide a continuous-discrete orthogonal decomposition of the additive group. We also introduce conditions of interval uniform asymptotic smoothness (along continuous directions) and pointwise uniform asymptotic smoothness condition (along discrete directions). Under these conditions, we establish the existence of traveling waves for discrete-time monotone systems. The obtained results are applied to a lattice-diffusion-impulse system on a cylinder with the Neumann boundary condition.</p>