We consider a class of evolution differential inclusions defining the so-called stop operator arising in elastoplasticity, ferromagnetism, and phase transitions. These differential inclusions depend on a constraint which is represented by a convex set that is called the characteristic set. For $\mathrm{BV}$ (bounded variation) data we compare different notions of $\mathrm{BV}$ solutions and study how the continuity properties of the solution operators are related to the characteristic set. In the finite-dimensional case...