  Suppose that a locally compact group $G$ acts freely and properly on the
right of a locally compact space $T$. Rieffel proved that if $\alpha$ is an
action of $G$ on a $C^*$-algebra $A$ and there is an equivariant embedding of
$C_0(T)$ in $M(A)$, then the action $\alpha$ of $G$ on $A$ is proper, and the
crossed product $A\rtimes_{\alpha,r}G$ is Morita equivalent to a generalised
fixed-point algebra $\Fix(A,\alpha)$ in $M(A)^\alpha$. We show that the
assignment $(A,\alpha)\mapsto\Fix(A,\alpha)$ extends to a functor $\Fix$ on a
category of $C^*$-dynamical systems in which the isomorphisms are Morita
equivalences, and that Rieffel's Morita equivalence implements a natural
isomorphism between a crossed-product functor and $\Fix$. From this, we deduce
naturality of Mansfield imprimitivity for crossed products by coactions,
improving results of Echterhoff-Kaliszewski-Quigg-Raeburn and Kaliszewski-Quigg
Raeburn, and naturality of a Morita equivalence for graph algebras due to
Kumjian and Pask.
