  Given a free additive convolution semigroup $\left(\mu_t\right)_{t\geq 0}$
and a probability measure $\nu$ on $\mathbb{R}$, we find the necessary and
sufficient conditions for the process $\mu_t \boxplus \nu$ 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 $\mu_t \boxplus \nu$
to be analytic at its zeros. These results are quantified by the L\'{e}vy
measure of the semigroup, making it fairly easy to construct many concrete
examples. Finally, we show that $\mu_t \boxplus \nu$ has a finite number of
connected components in its support if both the L\'{e}vy measure of
$\left(\mu_t\right)_{t \geq 0}$ and the initial law $\nu$ do.
