The spread of a particular trait in a cell population often is modelled by an
appropriate system of ordinary differential equations describing how the sizes
of subpopulations of the cells with the same genome change in time. On the
other hand, it is recognized that cells have their own vital dynamics and
mutations, leading to changes in their genome, mostly occurring during the cell
division at the end of its life cycle. In this context, the process is
described by a system of McKendrick type equations which resembles a network
transport problem. In this paper we show that, under an appropriate scaling of
the latter, these two descriptions are asymptotically equivalent
