  We study mixtures of free, monotone, and boolean independence described by
directed graphs (digraphs). For a sequence of digraphs $G_n = (V_n,E_n)$, we
give sufficient conditions for the limit $\widehat{\mu} = \lim_{n \to \infty}
\boxplus_{G_n}(\mu_n)$ to exist whenever the boolean convolution powers
$\mu_n^{\uplus |V_n|}$ converge to some $\mu$. This in particular includes
central limit and Poisson limit theorems, as well as limit theorems for each
classical domain of attraction. The hypothesis on the sequence of $G_n$ is that
the normalized counts of digraph homomorphisms from rooted trees into $G_n$
converge as $n \to \infty$, and we verify this for several families of examples
where the $G_n$'s converge in some sense to a continuum limit. In particular,
we obtain a new limit theorems for multiregular digraphs, as well as recovering
several limit theorems in prior work.
