  We extend the result of Lisini (Calc Var Partial Differ Equ 28:85-120, 2007)
on the superposition principle for absolutely continuous curves in
$p$-Wasserstein spaces to the special case of $p=1$. In contrast to the case of
$p>1$, it is not always possible to have lifts on absolutely continuous curves.
Therefore, one needs to relax the notion of a lift by considering curves of
bounded variation, or shortly BV-curves, and replace the metric speed by the
total variation measure. We prove that any BV-curve in a 1-Wasserstein space
can be represented by a probability measure on the space of BV-curves which
encodes the total variation measure of the Wasserstein curve. In particular,
when the curve is absolutely continuous, the result gives a lift concentrated
on BV-curves which also characterizes the metric speed. The main theorem is
then applied for the characterization of geodesics and the study of the
continuity equation in a discrete setting.
