Let $\mathcal{P}_2(\mathbb{R}^n)$ denote the set of all Borel probability measures on $\mathbb{R}^n$ with finite variance and weak topology. Let $X_t$ be a strong solution to the SDE with initial conditions $$ dX_t = \mu(t,X_t)dt + \sigma(t,X_t) dW_t, \mbox{ } X_0=x $$ for some Lipschitz-continuous functions $\mu$ and $\sigma$, and a Brownian motion $W_t$. Denote its conditional law $\nu_{t,x}:=\mathbb{P}(X_t \in \cdot|X_0=x)$.

Updated Question:

1. 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?
2. Fix $\Delta>0$ and suppose that $\mu$ and $\sigma$ do not depend on $t$, then does there exist a continuous function from $\mathcal{P}_2(\mathbb{R}^n)$ to itself, mapping $\nu_{x,t}$ to $\nu_{x,t+\Delta}$?