  This paper deals with the limit cases for $s$-fractional heat flows in a
cylindrical domain, with homogeneous Dirichlet boundary conditions, as $s\to
0^+$ and $s\to 1^-$\,.
  To this purpose, we describe the fractional heat flows as minimizing
movements of the corresponding Gagliardo seminorms, with respect to the $L^2$
metric. First, we provide an abstract stability result for minimizing movements
in Hilbert spaces, with respect to a sequence of $\Gamma$-converging uniformly
$\lambda$-convex energy functionals. Then, we provide the $\Gamma$-convergence
analysis of the $s$-Gagliardo seminorms as $s\to 0^+$ and $s\to 1^-$\,, and
apply the general stability result to such specific cases.
  As a consequence, we prove that $s$-fractional heat flows (suitably scaled in
time) converge to the standard heat flow as $s\to 1^-$, and to a degenerate ODE
type flow as $s\to 0^+$\,. Moreover, looking at the next order term in the
asymptotic expansion of the $s$-fractional Gagliardo seminorm, we show that
suitably forced $s$-fractional heat flows converge, as $s\to 0^+$\,, to the
parabolic flow of an energy functional that can be seen as a sort of
renormalized $0$-Gagliardo seminorm: the resulting parabolic equation involves
the first variation of such an energy, that can be understood as a zero (or
logarithmic) Laplacian.
