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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 negativevalues of the parameter $s=-\frac{\alpha}{2}$. Then, we determine the limitbehaviour of these gradient flows as $\alpha \to 0^+$ and $\alpha \to d^-$.  To this end we provide a preliminary $\Gamma$-convergence expansion for theRiesz interaction energy functionals. Then we apply abstract stability resultsfor uniformly $\lambda$-convex functionals which guarantee that$\Gamma$-convergence commutes with the gradient flow structure.

Parabolic $\alpha$-Riesz flows and limit cases $\alpha\to 0^+$, $\alpha\to d^-$