We present an abstract method for studying the stability of parabolic flows, exploiting the Gamma-convergence of the corresponding energy functionals. We apply such a result to analyse the behavior of the s-fractional heat flows, as s tends to 0 and to 1, and of the s-Riesz flows, as s tends to 0 and to d (where d is the dimension of the ambient space). Time permitting, we present also the corresponding stability results for the geometric flows.