Substitution Rule for Contour Integral of Meromorphic Function

Question

Given a meromorphic function on a domain $D\subset\mathbb{C}$. Is there a notion of substitution rule for a contour integral over the boundary $\partial{D}$? And what are the needed properties of the transformation function $g:D\rightarrow\tilde{D}$? I would suppose that one needs that $g$ is biholomorphic but even than I wouldn't know how to prove the statements. To me it's not clear why the sum of residues of the transformed integrand on the new domain yields the same as the sum of the residues of the old function on the old domain. Wouldn't one get terms which contain the derivatives of $g$ in the new residues? Thanks for a little hint.