This is the second part of a study of the nonlinear evolution of double-diffusive instabilities into laterally heated stably stratified liquids. Flows developing in initially doubly stratified systems are considered, i.e. in addition to stabilizing salinity distribution a destabilizing temperature distribution is present. Although the development of the intrusions is qualitatively similar to that in singly stratified liquids, import differences occur when the initial destabilizing temperature gradient becomes large. When the lateral heating is turned off, intrusions are still able to propagate. The main contribution of the paper is a detailed study of the physics of this self-propagation process.