Proper actions which are not saturated

If a locally compact group $G$ acts properly on a locally compact space $X$, then the induced action on $C_0(X)$ is proper in the sense of Rieffel, with generalised fixed-point algebra $C_0(G\backslash X)$. Rieffel's theory then gives a Morita equivalence between $C_0(G\backslash X)$ and an ideal $I$ in the crossed product $C_0(X)\times G$; we identify $I$ by describing the primitive ideals which contain it, and we deduce that $I=C_0(X)\times G$ if and only if $G$ acts freely. We show that if a discrete group $G$ acts on a directed graph $E$ and every vertex of $E$ has a finite stabiliser, then the induced action $\alpha$ of $G$ on the graph $C^*$-algebra $C^*(E)$ is proper. When $G$ acts freely on $E$, the generalised fixed-point algebra $C^*(E)^\alpha$ is isomorphic to $C^*(G\backslash E)$ and Morita equivalent to $C^*(E)\times G$, in parallel with the situation for free and proper actions on spaces, but this parallel does not seem to give useful predictions for nonfree actions.