We consider the free additive convolution semigroup {μ^t : t ≥ 1} and determine the local behavior of the density of μ^t at the endpoints and at any singular point of its support. We then study the free additive convolution of two multi-cut probability measures and show that its density decays either as a square root or as a cubic root at any endpoint of its support. The probability measures considered in this paper satisfy a power law behavior with exponents strictly between −1 and 1 at the endpoints of their supports.