Let $G$ be a Lie group acting on two smooth manifolds $M$ and $N$ such that both $M/G$ and $N/G$ are smooth manifolds. Suppose that $f: M\to N$ is a smooth proper $G$-map. Then it induces a smooth map $\bar f: M/G\to N/G$. My question is

Is the induced map $\bar f$ proper?

Here, proper means preimages of compact subsets are compact. All manifolds are assumed to be Hausdorff. Thanks advance.