Let me first explain the setup:

Let $(X_t)_{t \geq 0}$ be a stochastic process on some probability space $(\Omega,\mathcal{F},P)$ with values in a complete and separable metric space $E$ (e.g. $E = \mathbb{R}$) and let $\mathcal{G}$ be a sub-$\sigma$-field of $\mathcal{F}$. The conditional probability distribution $\mathcal{L}(X_t | \mathcal{G})$ can then be seen a as a random variable on $(\Omega,\mathcal{G})$ with values in $\mathcal{P}(E)$, which denotes the set of Borel probability measures on $E$. Then $\mathcal{P}(E)$ can be endowed with a metric that metrizes the weak convergence (also often called narrow convergence) of Borel probability measures on E, turning $\mathcal{P}(E)$ into a complete and separable metric space.

The question I have is about the regularization of the regular conditional probability distribution is the following:

If we now suppose that the paths $[0,\infty) \ni t \mapsto X_t(\omega) \in E$ are continuous for each $\omega \in \Omega$, can we choose versions/modifications of the collection of regular conditional probability distributions $(\mathcal{L}(X_t | \mathcal{G}))_{t \geq 0}$, such that the mappings $[0, \infty) \ni t \mapsto \mathcal{L}(X_t | \mathcal{G})(\omega) \in \mathcal{P}(E)$ become continuous for each $\omega \in \Omega$ ? Does this hold in this generality or do we need additional assumptions ?

If $(X_t)_{t \geq 0}$ is for example a Brownian motion, then the above desired regularisation would follow from Kolmogorov's continuity theorem. But I have not found any results for general continuous processes.

Thanks a lot in advance!