In this talk, I will discuss the transition from a compressible (inviscid) system with singular pressure towards a mixed compressible-incompressible system modeling partially congested dynamics. The two systems may be used for the modeling of mixtures, of collective motions, or partially free surface flows. From the mathematical point of view, I will present a first convergence result for small BV perturbations of a reference state represented by one or more partially congested propagating fronts.