In his 1998 work "Processes with free increments", Biane showed that Voiculescu's subordination property for free additive convolution can be significantly strengthened: the conditional expectation of the resolvent of a sum of two free random variables onto the algebra generated by one of the variables is itself a resolvent. Voiculescu extended this result to the context of tracial operator-valued probability spaces in "The Coalgebra of the Free Difference Quotient and Free Probability", offering a new conceptual framework which better explains the occurrence of the subordination phenomenon. In our talk, we explain how one may use a matrix construction to show that the subordination phenomenon, in an appropriately modified form, extends to the very general context of freeness over a Banach algebra. The talk is based on joint work with H. Bercovici, available at arXiv:2209.12710.