Do these two quantities behave the same?

Question

Let $\Phi_z(w)$ denote an involutive automorphism of the unit ball, $w,z\in \mathbb{B}^n$ and $f:\mathbb{B}^n\to \mathbb{C}$ be a holomorphic function. My question is the following: Let's take the gradient in terms of $w\in \mathbb{B}^n$ for the following function: $$\nabla f(\Phi_z(w)).$$ Fix $z\in \mathbb{B}$ for a while and take the supremum over $w\in \mathbb{B}^n$ of the norm of the gradient, that is: $$s=\sup_{w\in \mathbb{B}^n}|\nabla f(\Phi_z(w))|.$$ My question is the following: Is $s$ behaving similar to $|\nabla f(z)|$? Ideally, can I find constants $C_1,C_2$ such that: $$C_1 |\nabla f(z)|\leq s \leq C_2 |\nabla f(z)|?$$ My intuition tells me that the right inequality cannot be right in all cases. But if anyone has an answer I would be grateful. I am really stuck on this right now and don't know how to proceed.