  In this paper we introduce the notion of parabolic $\alpha$-Riesz flow, for
$\alpha\in(0,d)$, extending the notion of $s$-fractional heat flows to negative
values of the parameter $s=-\frac{\alpha}{2}$. Then, we determine the limit
behaviour of these gradient flows as $\alpha \to 0^+$ and $\alpha \to d^-$.
  To this end we provide a preliminary $\Gamma$-convergence expansion for the
Riesz interaction energy functionals. Then we apply abstract stability results
for uniformly $\lambda$-convex functionals which guarantee that
$\Gamma$-convergence commutes with the gradient flow structure.
