Action that is Bourbaki proper but not Palais proper

Let $M$ and $N$ be smooth finite dimensional manifolds with $M$ compact, and let $\dim(N)\ge \dim(M)$. Let $\text{Imm}(M,N)$ be the space of smooth immersions $N\to N$, which is a smooth manifold modelled on spaces $\Gamma(f^*TN)$ of smooth sections alonf immersions. Consider the regular Frechet Lie group $\text{Diff}(M)$ of all diffeomorphisms of $M$. See the Wikipedia article here for background.

In the paper here you find a more general version of the following result (there are some misprints but no mistake in the paper):

• The action $\text{Imm}(M,N)\times \text{Diff}(M)\to \text{Imm}(M,N)$ by composition from the right is smooth. The orbit space is Hausdorff in the quotient topology.

• On the open subspace $\text{Imm}_{free}(M,N)$ of free immersions the action of $\text{Diff}(M)$ is free (this is the definition of free) and is the action of smooth principal bundle.

• The isotropy group in $\text{Diff}(M)$ of a (non-free) immersion $f$ is always a finite group which acts strictly discontinuously on $M$, so that $f$ factors to a smooth manifold finitely covered by the isotropy group. Thus the orbit space $\text{Imm}(M,N)/\text{Diff}(M)$ is an infinite dimensional orbifold.

So this is a B-proper action, which is not P-proper since the group $\text{Diff}(M)$ is not locally compact - admittedly a trivial reason.

Here is another example: Consider a compact finite dimensional smooth manifold $M$, and let $\text{Met}(M)$ be the space of all smooth Riemannian metrics on $M$, an open subset in the Frechet space $\Gamma(S^2T^*M)$. Then the regular Frechet Lie group $\text{Diff}(M)$ of all diffeomorphisms of $M$ acts smoothly on $\text{Met}(M)$ by pullback. The quotient space $\text{Met}(M)/\text{Diff}(M)$ is Hausdorff in the quotient topology. The action is free on generic Riemannian metrics, and is the isometry group of a metric $g$ in general, which is a compact Lie group in general, since $M$ is compact. Thus this action is B-proper, but not P-proper, since the group $\text{Diff}(M)$ is not locally compact.