Total variation convergence in context of stochastic processes

Question

Given a stochastic process $(X_t)_{t\geq 0}$ on $(\Omega,\mathcal{F},\mathbb{P})$, with $\mu_t$ denoting the law of $X_t$ ($\mu_t=\mathbb{P}\circ X_t^{-1}$), the convergence in distribution $$X_t~\xrightarrow{d}~X_\infty$$ corresponds to $$\mu_t ~\xrightarrow{w}~ \mu_\infty,$$ (to be precise, convergence in $\sigma(\mathcal{M}(\Omega),C_b(\Omega))$-topology). Is there a similar correspondence for when $\mu_t\rightarrow\mu$ in total variation, i.e., when $\|\mu_\infty-\mu_t\|_{\mathrm{TV}}\rightarrow 0$? I am aware that on probability measures, (the restriction of) $(\mathcal{M}(\Omega),C_b(\Omega))$-topology coincides with the Dudley/dual bounded Lipschitz topology, but don't suppose there'd be a similar thing for the $\|\!\cdot\!\|_{\mathrm{TV}}$-topology (unless $\Omega$ is uniformly discrete). Note that I assume $\Omega$ to be a Polish space.