  Given two ideals $\mathcal{I}$ and $\mathcal{J}$ of holomorphic functions
such that $\mathcal{I} \subseteq \mathcal{J}$, we describe a comparison formula
relating the Andersson-Wulcan currents of $\mathcal{I}$ and $\mathcal{J}$. More
generally, this comparison formula holds for residue currents associated to two
generically exact Hermitian complexes together with a morphism between the
complexes.
  One application of the comparison formula is a generalization of the
transformation law for Coleff-Herrera products to Andersson-Wulcan currents of
Cohen-Macaulay ideals. We also use it to give an analytic proof by means of
residue currents of theorems of Hickel, Vasconcelos and Wiebe related to the
Jacobian ideal of a holomorphic mapping.
