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For simplicity take $E=\Bbb R$ and the time interval to be $[0,1]$, and think of $X=(X_t)_{0\le t\le 1}$ as a random element of $C=C([0,1]\to\Bbb R)$, a Polish space. We then have a regular conditional distribution of $X$ given $\mathcal G$, call it $Q=Q(\omega,B)$, $\omega\in\Omega, B\in\mathcal B(C)$. And the induced "marginal conditional distribution" $A\mapsto Q(\omega, \{\pi_t\in A\})$ will be weakly continuous in $t$. Here $\pi_t$, $0\le t\le 1$ are the coordinate projections from $C$ to $\Bbb R$.