  We study measure-valued solutions of the inhomogeneous continuity equation
$\partial_t \rho_t + {\rm div}(v\rho_t) = g \rho_t$ where the coefficients $v$
and $g$ are of low regularity. A new superposition principle is proven for
positive measure solutions and coefficients for which the recently-introduced
dynamic Hellinger-Kantorovich energy is finite. This principle gives a
decomposition of the solution into curves $t \mapsto h(t)\delta_{\gamma(t)}$
that satisfy the characteristic system $\dot \gamma(t) = v(t, \gamma(t))$,
$\dot h(t) = g(t, \gamma(t)) h(t)$ in an appropriate sense. In particular, it
provides a generalization of existing superposition principles to the
low-regularity case of $g$ where characteristics are not unique with respect to
$h$. Two applications of this principle are presented. First, uniqueness of
minimal total-variation solutions for the inhomogeneous continuity equation is
obtained if characteristics are unique up to their possible vanishing time.
Second, the extremal points of dynamic Hellinger-Kantorovich-type regularizers
are characterized. Such regularizers arise, e.g., in the context of dynamic
inverse problems and dynamic optimal transport.
