  In this article, using ideas of Liero, Mielke and Savar\'{e} in [21], we
establish a Kantorovich duality for generalized Wasserstein distances
$W_1^{a,b}$ on a generalized Polish metric space, introduced by Picolli and
Rossi. As a consequence, we give another proof that $W_1^{a,b}$ coincide with
flat metrics which is a main result of [25], and therefore we get a result of
independent interest that $\left(\mathcal{M}(X), W^{a,b}_1\right)$ is a
geodesic space for every Polish metric space $X$. We also prove that
$(\mathcal{M}^G(X),W_p^{a,b})$ is isometric isomorphism to
$(\mathcal{M}(X/G),W_p^{a,b})$ for isometric actions of a compact group $G$ on
a Polish metric space $X$; and several results of Gromov-Hausdorrf convergence
and equivariant Gromov-Hausdorff convergence of generalized Wasserstein spaces.
The latter results were proved for standard Wasserstein spaces in [22],[14] and
[8] respectively.
