  We extend the free convolution of Brown measures of $R$-diagonal elements
introduced by K\"{o}sters and Tikhomirov [Probab. Math. Statist. 38 (2018), no.
2, 359--384] to fractional powers. We then show how this fractional free
convolution arises naturally when studying the roots of random polynomials with
independent coefficients under repeated differentiation. When the proportion of
derivatives to the degree approaches one, we establish central limit
theorem-type behavior and discuss stable distributions.
