We study p-Wasserstein spaces over the branching spaces $\mathbb {R}^2$ and $[-1,1]^2$ equipped with the maximum norm metric. We show that these spaces are isometrically rigid for all $p\geq 1,$ meaning that all isometries of these spaces are induced by isometries of the underlying space via the push-forward operation. This is in contrast to the case of the Euclidean metric since with that distance the $2$-Wasserstein space over $\mathbb {R}^2$ is not rigid. Also, we highlight that the $1$-Wasserstein space is not rigid over the closed interval $[-1,1]$, while according to our result, its two-dimensional analog, the closed unit ball $[-1,1]^2$ with the more complicated geodesic structure is rigid.

We extend results proved by the second author (Amer. J. Math., 2009) for nonnegatively curved Alexandrov spaces to general compact Alexandrov spaces $X$ with curvature bounded below. The gradient flow of a geodesically convex functional on the quadratic Wasserstein space $\left( \mathcal{P}\left( X \right),\,{{W}_{2}} \right)$ satisfies the evolution variational inequality. Moreover, the gradient flow enjoys uniqueness and contractivity. These results are obtained by proving a first variation formula for the Wasserstein distance.