Maximal inequality regarding the empirical distribution function

Question

Let $F$ be a distribution function and $F_n$ the corresponding empirical distribution function. By the empirical process theory, we have the following stochastic equicontinuity condition : for some constant $C$, $$ \mathbb{E} \sup_{|t-s|<\delta} |F_n(t) - F(t) - (F_n(s) - F(s))| \leq C \sqrt{\delta/n} $$ Now let $\phi$ be a smooth function, how can we find out the order of the following object: $$ \mathbb{E} \sup_{|t-s|<\delta} |\phi(F_n(t)) - \phi(F(t)) - (\phi(F_n(s)) - \phi(F(s)))| $$ That is, is there any way to establish the stochastic equicontinuity condition after the transformation $\phi$?