Suppose we know that: if $X_n(0)=0$, then $X_n(\cdot)$ weakly converges to zero process in $D([0,\infty),\mathbb{R})$ as $n\rightarrow \infty$. Write this as: $$\mbox{When } X_n(0)=0, \,\,X_n(\cdot)\Rightarrow 0 \mbox{ in } D([0,\infty),\mathbb{R}) \mbox{ as }n\rightarrow \infty.$$
Now, suppose $X_n(0)\rightarrow 0$ in probability. Does it still hold $$X_n(\cdot)\Rightarrow 0 \mbox{ as }n\rightarrow \infty?$$ Any help or insights would be greatly appreciated.
First let $Y_n(t)=X_n(t)-X_n(0)$ for all $t\geq 0$ and $n\in\mathbb{N}$. Since $Y_n(0)=0$ your assumption allows us to conclude that $Y_n \Rightarrow_n 0$.
Now let $\rho$ denote the Skorokhod metric on $D([0,\infty),\mathbb{R})$. If you can show that that $\rho(X_n,Y_n) \stackrel{P}{\to} 0$ as $n\to\infty$, then a Slutsky'ish theorem (Theorem 3.1; Convergence of probability measures - P. Billingsley) yields that $X_n \Rightarrow 0$.