  We start with a random polynomial $P^{N}$ of degree $N$ with independent
coefficients and consider a new polynomial $P_{t}^{N}$ obtained by repeated
applications of a fraction differential operator of the form $z^{a}%
(d/dz)^{b},$ where $a$ and $b$ are real numbers. When $b>0,$ we compute the
limiting root distribution $\mu_{t}$ of $P_{t}^{N}$ as $N\rightarrow\infty.$ We
show that $\mu_{t}$ is the push-forward of the limiting root distribution of
$P^{N}$ under a transport map $T_{t}$. The map $T_{t}$ is defined by flowing
along the characteristic curves of the PDE satisfied by the log potential of
$\mu_{t}.$ In the special case of repeated differentiation, our results may be
interpreted as saying that the roots evolve radially \textit{with constant
speed} until they hit the origin, at which point, they cease to exist. For
general $a$ and $b,$ the transport map $T_{t}$ has a free probability
interpretation as multiplication of an $R$-diagonal operator by an $R$-diagonal
"transport operator." As an application, we obtain a push-forward
characterization of the free self-convolution semigroup $\oplus$ of radial
measures on $\mathbb{C}$.
  We also consider the case $b<0,$ which includes the case of repeated
integration. More complicated behavior of the roots can occur in this case.
