  We prove that the system of subordination equations, defining the free
additive convolution of two probability measures, is stable away from the edges
of the support and blow-up singularities by showing that the recent smoothness
condition of Kargin is always satisfied. As an application, we consider the
local spectral statistics of the random matrix ensemble $A+UBU^*$, where $U$ is
a Haar distributed random unitary or orthogonal matrix, and $A$ and $B$ are
deterministic matrices. In the bulk regime, we prove that the empirical
spectral distribution of $A+UBU^*$ concentrates around the free additive
convolution of the spectral distributions of $A$ and $B$ on scales down to
$N^{-2/3}$.
