The dynamical properties of image-potential states on stepped Cu(117) and Cu(118) surfaces were studied by time- and angle-resolved two-photon photoelectron spectroscopy. Interaction with the step-induced potential leads to quasielastic anisotropic scattering between image-potential-state bands. In particular resonant interband scattering from image-potential states with quantum numbers n>=2 to the n=1 band and quasielastic intraband scattering within the n=1 band show high efficiency. In spite of the higher step density of Cu(117) resonant scattering is about four times larger on Cu(118). This distinction is attributed to the different step distributions of both surfaces and to the concomitant correlation of the step arrangement on short- and long-range scales which was studied by scanning tunneling microscopy. Details of the surface band structure lead to different boundary conditions and may weakly affect scattering probabilities.