Weak convergence $B\circ f_n\to B\circ f$

Question

Let $B$ be a brownian motion and consider a sequence of continuous functions $f_n$ defined on $[0,1]$, such that $f_n(x)\to f(x)$ for each $x$. Is it true that $B\circ f_n$ converges to $B\circ f$ as a process? I think this must hold but I do not find a direct way to prove it. I would appreciate any idea on this. Thank you.