The following discussion is based on the book Gradient Flows by Ambrosio, Gigli, and Savare (2008).

Consider $p$-Wasserstein distance with $p>1$ (for the sake of uniqueness). Let $\gamma$ be the optimal transport plan between $\mu$ and $\nu$ under the $p$-Wasserstein distance. Denote by $\pi^i$ be the $i$-th projection ($i=1,2$) and by $\#$ the pushforward of measures (see Section 5.2). Define $\lambda_\alpha = ((1-\alpha) \pi^1 + \alpha \pi^2)_\# \gamma$. Then $W_p(\mu,\lambda_\alpha)=\alpha W_p(\mu,\nu)$, $W_p(\lambda_\alpha,\nu)=(1-\alpha)W_p(\mu,\nu)$ and $W_p(\mu,\lambda_\alpha) + W_p(\lambda_\alpha,\nu)=W_p(\mu,\nu)$.

The following discussion is based on the book Gradient Flows by Ambrosio, Gigli, and Savare (2008).

Consider $p$-Wasserstein distance with $p>1$ on a Hilbert space (for the sake of uniqueness). Let $\gamma$ be the optimal transport plan between $\mu$ and $\nu$ under the $p$-Wasserstein distance. Denote by $\pi^i$ be the $i$-th projection ($i=1,2$) and by $\#$ the pushforward of measures (see Section 5.2). Define $\lambda_\alpha = ((1-\alpha) \pi^1 + \alpha \pi^2)_\# \gamma$. Then according to Section 7.2, we have $W_p(\mu,\lambda_\alpha)=\alpha W_p(\mu,\nu)$, $W_p(\lambda_\alpha,\nu)=(1-\alpha)W_p(\mu,\nu)$ and $W_p(\mu,\lambda_\alpha) + W_p(\lambda_\alpha,\nu)=W_p(\mu,\nu)$.