We consider the free additive convolution semigroup $\lbrace μ^{\boxplus t}:\,t\ge 1\rbrace$ and determine the local behavior of the density of $μ^{\boxplus t}$ 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 endpoints 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.