Please consider the integral \[ F(s) = \int_{c + s - i \infty}^{c + s + i\infty} f(z, s) \, dz \] for $s = a + bi$.
I am sure that $F(a + bi)$ is invariant when the real part of $s$ is taken out from the path limits.
What will happen when $bi$ is taken out from the path limits? If the integrand is independent of $s$ then $F(s)$ would not be affected. But for the integrand dependent of $s$, I can't be pretty sure.
If we take $s$ in path limits as a sort of "shift" in path, then $F(s)$ seems to be constant for all $s$.