The objective of this work is a numerical study of the stability properties and the evolution of the eastward-travelling baroclinic modons – coherent vortex structures specific to stratified geophysical fluids where differential rotation (the β-effect) is of the essence. In the vortices under study, the initial dependence of the potential vorticity (PV) upon the streamfunction is piecewise-linear, the barotropic component is dipolar, the baroclinic component is circularly symmetric about the vertical axis, and the boundary of the trapped-fluid region (in which the vorticity contours are closed) is a circular cylinder. These modons are shown to be stable for a wide range of parameters. In two- and three-layer fluids, modons of this type are shown to be able to transit to even more durable states, in which the trapped-fluid area is oval in shape and the PV versus streamfunction dependence in this domain is nonlinear. Possible transition mechanisms and linkage between the circular and oval modons are discussed.