(Rate of) Convergence in distribution and Laplace transform of random variables/stochastic processes

Question

Let $X_t^n$ and $X_t$ be stochastic processes (with finite moments), and assume that for every $t>0$, $\lambda>0$ and bounded continuous function $\varphi$, \begin{equation} \int_0^te^{-\lambda s}E\varphi(X_s^n)d s\to\int_0^te^{-\lambda s}E\varphi(X_s)d s,\quad as \quad n\to\infty. \end{equation} Can we conclude that for every $t>0$, $$ E\varphi(X_t^n)\to E\varphi(X_t),\quad as \quad n\to\infty ? $$ A further question is that, if we know the rate of convergence, i.e., for a constant $C_0>0$, $$ \bigg|\int_0^te^{-\lambda s}E\varphi(X_s^n)d s-\int_0^te^{-\lambda s}E\varphi(X_s)d s\bigg|\leq C_0\frac{1}{n}, $$ can we get $$ \big|E\varphi(X_t^n)\to E\varphi(X_t)\big|\leq C_1\frac{1}{n} ? $$ Thanks for your time and answer. Any comments or suggestions are welcome.