We investigate the scaling behavior for roughening and coarsening of mounds during unstable epitaxial growth. By using kinetic Monte Carlo (KMC) simulations of two lattice-gas models of crystal surfaces, we find scaling exponents that characterize roughening and coarsening at long times. Our simulation data show that these exponents have a complicated dependence on key model parameters that describe a step edge barrier and downward transport mechanisms. This behavior has not been fully described in previous works. In particular, we find that these scaling exponents vary continuously with parameters controlling the surface current. The kinetic processes of the KMC models that we employ include surface diffusion, edge diffusion, step-edge barriers, and also account for transient kinetics during deposition via downward funneling and transient mobility. Our extensive simulations make evident the salient interplay between step-edge barrier strength and transient kinetic processes.