Suppose you have a Lie group $G$ acting properly on a manifold $M$. One can show that the space of orbits $M/G$ is a second-countable, locally compact and Hausdorff. Let $\pi: M\rightarrow M/G$ be the projection.
One can construct a bounded, smooth function (sometimes called a "cut-off function") $$c:M\rightarrow [0,\infty),$$ with two properties:
For any subset $L\subseteq M$ such that $\pi(L)$ is compact in $M/G$, the intersection of $L$ with the support of $c$ is compact (or its closure is).
$\int_G c(g^{-1}x) dg = 1$ for all $x\in M$.
An example of a construction is given on page 8 of https://arxiv.org/pdf/1608.06375.pdf.
I'd like to know: given the above set-up, is it always possible to construct such a $c$ so that the norm of the differential $|dc|$ fades to $0$ at infinity, or where $c$ is a limit of such functions in the space of bounded continuous functions $C_b(M)$ with the sup-norm?
(By "fading to $0$ at infinity", I mean that for any $\epsilon > 0$, there exists a compact $K\subseteq M$ such that on $M\backslash K$, $|dc|$ is uniformly less than $\epsilon$?)