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 $N^{-2/3}$.