  The Wasserstein distances $W_p$ ($p\geq 1$), defined in terms of solution to
the Monge-Kantorovich problem, are known to be a useful tool to investigate
transport equations. In particular, the Benamou-Brenier formula characterizes
the square of the Wasserstein distance $W_2$ as the infimum of the kinetic
energy, or action functional, of all vector fields moving one measure to the
other.
  Another important property of the Wasserstein distances is the
Kantorovich-Rubinstein duality stating the equality between the distance $W_1$
and the supremum of the integrals of Lipschitz continuous functions with
Lipschitz constant bounded by one.
  An intrinsic limitation of Wasserstein distances is the fact that they are
defined only between measures having the same mass. To overcome such
limitation, we recently introduced the generalized Wasserstein distances
$W_p^{a,b}$, defined in terms of both the classical Wasserstein distance $W_p$
and the total variation (or $L^1$) distance. Here $p$ plays the same role as
for the classic Wasserstein distance, while $a$ and $b$ are weights for the
transport and the total variation term.
  In this paper we prove two important properties of the generalized
Wasserstein distances:
  1) a generalized Benamou-Brenier formula providing the equality between
$W_2^{a,b}$ and the supremum of an action functional, which includes a
transport term (kinetic energy) and a source term.
  2) a duality \`{a} la Kantorovich-Rubinstein establishing the equality
between $W_1^{1,1}$ and the flat metric.
