Rippled one-dimensionally periodic structures are commonly seen in the experimental studies of the epitaxial growth and erosion on low symmetry rectangular (110) crystal surfaces. Rippled states period (wavelength) and amplitude grow via a coarsening process that involves motion and annihilations of the dislocations disordering perfect periodicity of these structures. Unlike the ordinary dislocations in equilibrium systems, the dislocations of the growing rippled states are genuinely traveling objects, never at rest. Here, we theoretically elucidate the structure and dynamics of these far-from-equilibrium topological defects. We derive fundamental dislocation dynamics laws that relate the dislocation velocity to the rippled state period. Next, we use our dislocations velocity laws to derive the coarsening laws for the temporal evolution of the rippled state period lambda and the ripple amplitude w (surface roughness). For the simple rippled states on (110) surfaces, we obtain the coarsening law lambda approximately w approximately t{2/7} . Under some circumstances however, we find that these states may exhibit a faster coarsening with lambda approximately w approximately t{1/3} . We also discuss the dislocations in the rectangular rippled surface states for which we derive the coarsening law with lambda approximately w approximately t{1/4} . The coarsening laws that occur at the transition from the rippled to the rhomboidal pyramid state are also discussed, as well as the crossover effects that occur in rippled states in the proximity of this transition on (110) crystal surfaces.