  In [Gwiazda, Jamr\'oz, Marciniak-Czochra 2012] a framework for studying cell
differentiation processes based on measure-valued solutions of transport
equations was introduced. Under application of the so-called
measure-transmission conditions it enabled to describe processes involving both
discrete and continuous transitions. This framework, however, admits solutions
which lack continuity with respect to initial data. In this paper, we modify
the framework from [Gwiazda, Jamr\'oz, Marciniak-Czochra 2012] by replacing the
flat metric, known also as bounded Lipschitz distance, by a new
Wasserstein-type metric. We prove, that the new metric provides stability of
solutions with respect to perturbations of initial data while preserving their
continuity in time. The stability result is important for numerical
applications.
