Let $\mu_n,n\in \mathbb N$ be a random probability measures and let $\mu$ be a deterministic probability measure on $\mathbb R$. That is to say, that the $\mu_n$ are measurable maps from a probability space $(\Omega,\mathcal{T},\mathbf{P})$ to the space of $M_1(\mathbb R)$ equipped with the Borel-$\sigma$-algebra generated by the topology of weak convergence. Assume that the expected measures $\nu_n:=\mathbf E\mu_n$, defined via duality as $\int f~d\nu_n:=\mathbf E\int f~d\mu_n$ for all $f\in C_b(\mathbb R)$, converge weakly to $\mu$, i.e. that for all $f\in C_b(\mathbb R)$ we have the convergence of $\mathbf E\int f d\mu_n$ to $\int f d\mu$. By Levy's continuity Theorem this is equivalent to the statement that the characteristic function $\phi_{\nu_n}(t):=\int e^{itx}~d\nu_n(x)$ converges pointwise to the characteristic function $\phi_\mu$ of $\mu$. Is a similar statement also true for *weak convergence in probability* the random measures, i.e. is there a connection between the following statements?

For all $\epsilon>0$ the sequence $\mathbf{P}(d_{BL}(\mu_n,\mu)>\epsilon)$ converges to zero, where $d_{BL}$ is the bounded Lipschitz metric $$d_{BL}(\mu,\nu)=\sup\left\{\left\lvert\int f~d\mu-\int f~d\nu\right\lvert~:~f\colon\mathbb{R}\to\mathbb{R}\text{ is 1-Lipschitz and }\lVert f\lVert_\infty\leq1\right\}$$ (which completely metrizes the topology of weak convergence)

For all $f\in C_b(\mathbb R)$ it holds that $\int f~d\mu_n$ converges in probability to $\int f~d\mu$.

For all $t\in \mathbb R$, $\phi_{\mu_n}(t)$ converges to $\phi_\mu(t)$ in probability.

On some small interval $[0,\epsilon]$, the sequence $\sup_{0\leq t\leq \epsilon}\lvert \phi_{\mu_n}(t)-\phi_\mu(t)\lvert$ converges in probability to zero.

For all $k\in\mathbb N$, the sequence of moments $\int x^k~d\mu_n$ converges in probability to $\int x^k d\mu$.

For all $k\in\mathbb N$, the sequence of expected moments $\mathbf E\int x^k~d\mu_n$ converges to $\int x^k d\mu$ and the variances $\mathbf{Var}\int x^k~d\mu_n$ converge to zero.

Clearly, for the last two statements to be equivalent to any of the above, we would need some condition on the determinacy of $\mu$ via its moments. Therefore, for simplicity assume that $\mu$ is subgaussian, and that the $\mu_n$ are uniformly subgaussian in the sense that $\mathbf E\mu_n([-R,R]^c)\leq C e^{-C R^2}$ for some $C$ and all $n,R$.