Other than point masses in a length space or Gaussian measures in $\mathbb{R}^n$, I don't know of examples of families which are totally geodesic in Wasserstein geometry.

However, it is possible to take a smooth family of probability measures and induce it with a Riemannian metric coming from the Wasserstein geometry. This idea is sometimes known as "Wasserstein Information Geometry" and was studied by Wuchen Li and Jiaxi Zhao [1].
The general framework is to consider a smooth parametrized family $\mathcal{F}$ of probability measures and assume the underlying sample space $S$ has some metric structure. In several important cases (e.g., when $S$ is a finite set or a Riemannian manifold), the 2-Wasserstein space $\mathbb{P}_2(S)$ admits a formal Riemannian structure known as the Otto metric. As such, $\mathcal{F}$ can be understood as a sub-manifold of a "Riemannian manifold." (I put "Riemannian manifold" in quotes here because $\mathbb{P}_2(S)$ is generally not a Banach manifold.)

The same way that a surface in $\mathbb{R}^3$ inherits a Riemannian metric from the ambient Euclidean space, we can induce $\mathcal{F}$ with a Riemannian metric from the ambient Otto metric, and in most cases this will be a genuine Riemannian metric (even though $\mathbb{P}_2(S)$ will often be singular). In general, $\mathcal{F}$ will not be a totally geodesic submanifold of $\mathbb{P}_2(S)$, so you cannot compute distance between points in $\mathcal{F}$ in terms of their distances in $\mathbb{P}_2(S)$ or vice-versa.

*Li, Wuchen; Zhao, Jiaxi*, **Wasserstein information matrix**, Inf. Geom. 6, No. 1, 203-255 (2023). ZBL1521.94014.