  We consider a rate-independent system with nonconvex energy under
discontinuous external loading. The underlying space is finite dimensional and
the loads are functions in $BV([0,T];\mathbb{R}^d)$. We investigate the
stability of various solution concepts w.r.t. a sequence of loads converging
weakly$*$ in $BV([0,T];\mathbb{R}^d)$ with a particular emphasis on the
so-called normalized, $\mathfrak{p}$-parametrized balanced viscosity solutions.
By means of two counterexamples, it is shown that common solution concepts are
not stable w.r.t. weak$*$ convergence of loads in the sense that a limit of a
sequence of solutions associated with these loads need not be a solution
corresponding to the load in the limit. We moreover introduce a new solution
concept, which is stable in this sense, but our examples show that this concept
necessarily allows "solutions" that are physically meaningless.
