Suppose we have two absolutely continuous distribution functions $F$ and $G$ with densities $f$ and $g,$ respectively. Assuming a quadratic cost function, the Wasserstein geodesic at time $t$ is the measure $\mu_t$ with quantile function $H_t^{-1} = (1-t)F^{-1} + t G^{-1}.$ See Section 2.1 here for a discussion. I'm interested in calculating the density $h_t$ associated with the geodesic at time $t$. My strategy is to observe that if $U \sim~ \text{Unif}(0,1)$ then $Z_t = H_t^{-1}(U) := \varphi_t(U) \sim \mu_t.$ Thus, we should be able to calculate $h_t$ by a change of variables. (I introduced the $\varphi_t$ notation because otherwise calculating the derivatives of inverses can get very confusing.)
Letting $u$ be the density of the uniform distribution on the unit interval, here's my attempt:
\begin{align*} h_t(z) &= u\left[\varphi_t^{-1}(z)\right]\times \frac{d}{dz}\varphi_t^{-1}(z)\\ &=\frac{d}{dz}\varphi_t^{-1}(z) \\ &= \frac{1}{\varphi_t'\left[\varphi_t^{-1}(z)\right]}. \end{align*}
Then we have that \begin{align*} \varphi_t'(x) &= (1-t)\frac{d}{dz} F^{-1}(x) + t\frac{d}{dz} G^{-1}(x)\\ &= \frac{1-t}{f[F^{-1}(x)]} + \frac{t}{g[G^{-1}(x)]}. \end{align*}
Substituting this expression for $\varphi_t'$ into our previous expression for $h,$ we have
\begin{align*} h_t(z) &= \frac{1}{ \frac{1-t}{f\left\{F^{-1}\left[\varphi_t^{-1}(z)\right]\right\}} + \frac{t}{g\left\{G^{-1}\left[\varphi_t^{-1}(z)\right]\right\}} } \\ &= \frac{f\left\{F^{-1}\left[\varphi_t^{-1}(z)\right]\right\}g\left\{G^{-1}\left[\varphi_t^{-1}(z)\right]\right\}}{(1-t)g\left\{G^{-1}\left[\varphi_t^{-1}(z)\right]\right\} + tf\left\{F^{-1}\left[\varphi_t^{-1}(z)\right]\right\}}\\ &=\frac{f\left\{F^{-1}\left[H_t(z)\right]\right\}g\left\{G^{-1}\left[H_t(z)\right]\right\}}{(1-t)g\left\{G^{-1}\left[H_t(z)\right]\right\} + tf\left\{F^{-1}\left[H_t(z)\right]\right\}}. \end{align*}
Does this derivation seem correct? I assume there is rarely a nice expression for $H_t$ and it must be calculated numerically instead. Does this appear somewhere in the literature?