Let $\mathcal{P}(\mathbb{R}^n)$ denote the set of all Borel probability measures on $\mathbb{R}^n$ with the weak topology. Let $X_t$ be a strong solution to the SDE with initial conditions $$ dX_t = \mu_tdt + \sigma_t dW_t, \mbox{ } X_0=x $$ for some Lipschitz-continuous functions $\mu$ and $\sigma$, and a Brownian motion $W_t$. Denote its conditional law $\nu_{t,x}:=\mathbb{P}(X_t \in \cdot|X_0=x)$.
Is the map $(x,t)\mapsto \nu_{t,x}$ from $\mathbb{R}^n\times [0,\infty)$ to $\mathcal{P}_2(\mathbb{R}^n)$ ever continuous?
What I've tried: I tried to use the Feynman-Kac formula or approach the problem somehow using disintegration, but it doesn't seem to be clear how to go about it; or even if the "result" is true..