  We consider a fixed free and proper action of a locally compact group $G$ on
a space $T$, and actions $\alpha:G\to \Aut A$ on $C^*$-algebras for which there
is an equivariant embedding of $(C_0(T),\rt)$ in $(M(A),\alpha)$. A recent
theorem of Rieffel implies that $\alpha$ is proper and saturated with respect
to the subalgebra $C_0(T)AC_0(T)$ of $A$, so that his general theory of proper
actions gives a Morita equivalence between $A\rtimes_{\alpha,r}G$ and a
generalised fixed-point algebra $A^\alpha$. Here we investigate the functor
$(A,\alpha)\mapsto A^\alpha$ and the naturality of Rieffel's Morita
equivalence, focusing in particular on the relationship between the different
functors associated to subgroups and quotients. We then use the results to
study induced representations for crossed products by coactions of homogeneous
spaces $G/H$ of $G$, which were previously shown by an Huef and Raeburn to be
fixed-point algebras for the dual action of $H$ on the crossed product by $G$.
