  Given two polynomials $p(x), q(x)$ of degree $d$, we give a combinatorial
formula for the finite free cumulants of $p(x)\boxtimes_d q(x)$. We show that
this formula admits a topological expansion in terms of non-crossing
multi-annular permutations on surfaces of different genera.
  This topological expansion, on the one hand, deepens the connection between
the theories of finite free probability and free probability, and in particular
proves that $\boxtimes_d$ converges to $\boxtimes$ as $d$ goes to infinity. On
the other hand, borrowing tools from the theory of second order freeness, we
use our expansion to study the infinitesimal distribution of certain families
of polynomials which include Hermite and Laguerre, and draw some connections
with the theory of infinitesimal distributions for real random matrices.
  Finally, building off our results we give a new short and conceptual proof of
a recent result [Steinerberger (2020), Hoskins and Kabluchko (2020)] that
connects root distributions of polynomial derivatives with free fractional
convolution powers.
