It turns out that for most convex domains $\Omega$, one can find smooth probability measures $\rho_0$ and $\rho_2$ which are supported on $\Omega$, have strictly positive density everywhere on $\Omega$, but whose barycenter (in the $2$-Wasserstein sense) has non-convex support (so has support strictly smaller than $\Omega$). This phenomena was recently discovered by Santambrogio and Wang [1]. As such, I'm not sure of sufficient conditions to ensure that a lower bound on the density holds throughout the displacement interpolation. To the best of my knowledge, this question is still open.
One follow-up question is whether it is possible to establish lower bounds for $d \rho_t$ on its support. The densities of $\rho_0$ and $\rho_1$ do not go to zero at the boundary $\partial \Omega$, so one might imagine that this continues to be the case for $\rho_t$ (although the support is evolving). By contrast, it might also be the case that $d \rho_t$ vanishes continuously at some points within the domain. I have no idea which of these two phenomena occur (perhaps both are possible?) and I think it would be very interesting to find out.
[1] Santambrogio, Filippo; Wang, Xu-Jia, Convexity of the support of the displacement interpolation: counterexamples, Appl. Math. Lett. 58, 152-158 (2016). ZBL1345.49056.