We consider a one-dimensional transport (balance) equation with velocity which has non-Lipschitz zeroes. This leads to non-uniqueness and concentration of characterics and dynamics with both discrete and continuous components. To deal with these effects, we use measure-valued solutions and the so-called measure-transmission conditions. A metric in the space of Radon measures allowing to define unique and stable solutions is introduced. The equation under consideration was proposed as a structured population model of cell differentiation.