  We study subordination of free convolutions. We prove that for free random
variables $X,Y$ and a Borel function $f$ the conditional expectation
$E_\varphi\left[ (z-X-f(X)Yf^*(X))^{-1}| X\right]$, is a resolvent again. This
result allows explicit calculation of the distribution of $X+f(X)Yf^*(X)$. The
main tool is a formula for conditional expectations in terms of Boolean
cumulant transforms, generalizing subordination formulas for free additive and
multiplicative convolutions.
