Given a free additive convolution semigroup $\left(μ_t\right)_{t\geq 0}$ and a probability measure $ν$ on $\mathbb{R}$, we find the necessary and sufficient conditions for the process $μ_t \boxplus ν$ to be Lebesgue absolutely continuous with a positive and analytic density throughout $\mathbb{R}$ at all time $t>0$. For semigroups without this property, we find the necessary and sufficient conditions for the density of $μ_t \boxplus ν$ to be analytic at its zeros. These results are quantified by the Lévy measure of the semigroup, making it fairly easy to construct many concrete examples. Finally, we show that $μ_t \boxplus ν$ has a finite number of connected components in its support if both the Lévy measure of $\left(μ_t\right)_{t \geq 0}$ and the initial law $ν$ do.