  We study a family of nonlinear integral flows that involve Riesz potentials
on Riemannian manifolds. In the Hardy-Littlewood-Sobolev (HLS) subcritical
regime, we present a precise blow-up profile exhibited by the flows. In the HLS
critical regime, by introducing a \textit{dual $Q$ curvature} we demonstrate
the concentration-compactness phenomenon. If, in addition, the integral kernel
matches with the Green's function of a conformally invariant elliptic operator,
this critical flow can be considered as a dual Yamabe flow. Convergence is then
established on the unit spheres, which is also valid on certain locally
conformally flat manifolds.
