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Two and three-layer models of stratified flows in hydrostatic balance are studied. For the former,nonlinear transformations are found that map [baroclinic] two-layer flows with either rigid top andbottom lids or vertical periodicity, into [barotropic] single-layer, shallow water free-surface flows. Wehave previously shown that two-layer flows with Richardson number greater than one are non-linearlystable, in the following sense: when the system is well-posed at a given time, it remains well-posedthrough the nonlinear evolution. Here, we give a general necessary condition for the nonlinearstability of systems of mixed type. For three-layer flows with vertical periodicity, the domains oflocal stability are determined and the system is shown not to satisfy the necessary condition fornonlinear stability. This means that there are wave-motions that evolve into into shear unstableflows.