  We classify spin ALE ancient Ricci flows and spin ALE expanding solitons with
suitable groups at infinity. In particular, the only spin ancient Ricci flows
with groups at infinity in $SU(2)$ and mild decay at infinity are hyperk\"ahler
ALE metrics. The main idea of the proof, of independent interest, consists in
showing that the large-scale behavior of Perelman's $\mu$-functional on any ALE
orbifold with non-negative scalar curvature is controlled by a renormalized
$\lambda_{\mathrm{ALE}}$-functional related to a notion of weighted mass.
