Let $K$ be a not differentiable kernel, $\sigma$ a (at least) $C^1$ function, Lipschitz of constant $L > 0$, $X_s, Y_s$ two cadlag processes.
Let $f(t,s,X_{\cdot})$ a measurable function and indicate with $f_1$ the derivate of $f$ in the first variable ($t$). Can we say that $$\sup_{u<s}\left|f_1(t,u,X_{\cdot})-f_1(t,u,Y_{\cdot}) \right|\leq C \sup_{u<s}|X_s-Y_s|$$ for some constant $C$?
Is there a way to exploit the regularity of $\sigma$ and bypass the not differentiability of $K$?