In this paper we continue our study of the Kelvin-Helmholtz (KH) instability in relativistic planar jets following the long-term evolution of the numerical simulations which were introduced in Paper I. The models have been classified into four classes (I to IV) with regard to their evolution in the nonlinear phase, characterized by the process of jet/ambient mixing and momentum transfer. Models undergoing qualitatively different non-linear evolution are clearly grouped in well-separated regions in a jet Lorentz factor/jet-to-ambient enthalpy diagram. Jets with a low Lorentz factor and small enthalpy ratio are disrupted by a strong shock after saturation. Those with a large Lorentz factor and enthalpy ratio are unstable although the process of mixing and momentum exchange proceeds to a longer time scale due to a steady conversion of kinetic to internal energy in the jet. In these cases, the high value of the initial Lorentz seems to prevent transversal velocity from growing far enough to generate the strong shock that breaks the slower jets. Finally, jets with either high Lorentz factors and small enthalpy ratios or low Lorentz factors and large enthalpy ratios appear as the most stable. In the long term, all the models develop a distinct transversal structure (shear/transition layers) as a consequence of KH perturbation growth. The properties of these shear layers are analyzed in connection with the parameters of the original jet models.