This question is answered in comments [fedja][1]. Consider 
$$
f(t,v)=\min_s \sqrt{(X(s)-v)^2+(t-s)^2}
$$ 
This function is Lipschitz, as the distance from $(t,v)$ to the graph of $X$. Whether or not $X$ has bounded variations, $f(t,X(t))$ is identically zero.


  [1]: https://mathoverflow.net/users/1131/fedja