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 $$ dX_t = \mu_tdt + \sigma_t dW_t, \mbox{ } X_0=x $$ for some continuous functions $\mu$ and $\sigma$, and a Brownian motion $W_t$. Let $\nu_t$ denote the law of this process at time $t$. Under what conditions does there exist a continous function $f:\mathbb{R}^n\times [0,\infty)\rightarrow\mathcal{P}_2(\mathbb{R}^n)$ mapping $\nu_s$ to $\nu_t$ for each $s\leq t$?

Edit: Based on the comment, can we do this using the Feynman-Kac forumla?