TL;DR Usually, RG flow involves scaling the metric $g_{\mu\nu}\to\lambda^2 g_{\mu\nu}$ and seeing how couplings change. But if we're quantising $g_{\mu\nu}$ itself, I don't see how this process can be well-defined.
RG Recap
Let me first recap how I understand RG flow to work when the metric is fixed. For simplicity, consider a scalar field $\phi$ on a Euclidean $d$-dimensional space. I will also massively simplify things by assuming the theory is finite: the path integral converges without any need for a cutoff. Some may feel this removes the need to discuss renormalisation, but I disagree: we can still ask how physics changes as we "zoom out", which is exactly what the RG flow tells us. True, if the path integral converges, the RG flow is simple and boring (i.e. couplings just scale according to their engineering dimension), but hopefully this simplicity should make my issue clearer.
Let $M$ be a compact region. Given a metric $g_{\mu\nu}$ and couplings $\mathbf{c} = (c_0,c_1,...)$, the amplitude for a boundary state $\phi_{\partial M}$ is:
\begin{align} Z[\phi_{\partial M}; g_{\mu\nu},\mathbf{c}]= \int_{\phi|_{\partial M}=\phi_{\partial M}} D\phi \exp(-S[\phi; g_{\mu\nu},\mathbf{c}]), \end{align}
where the action $S$ is given by \begin{align} S[\phi; g_{\mu\nu},\mathbf{c}] = \int d^4x \sqrt{g} \left( -g^{\mu\nu}\nabla_\mu \phi \nabla_\nu\phi + \sum c_n\phi^n \right). \end{align}
Let's just ignore the fact that this theory isn't finite, and proceed formally. We'd like to know how the physics changes as we zoom out, i.e. as we scale distances as $g_{\mu\nu}\to \lambda^2 g_{\mu\nu}$. For any $\lambda\neq 0$, we observe that \begin{align} S[\phi;g_{\mu\nu},\mathbf{c}] = S[\lambda^{-1} \phi;\lambda^2g_{\mu\nu},\mathbf{c}'] \end{align} with new couplings given by $c_k' = \lambda^{n-4} c_k$. (In reality there'd be quantum corrections to these new couplings, due to the presence of a cutoff, but we're ignoring this...). Through careful relabelling of variables in the path integral, we arrive at: \begin{align} Z[\phi_{\partial M}; g_{\mu\nu},\mathbf{c}] = Z[\lambda\phi_{\partial M}; \lambda^2 g_{\mu\nu},\mathbf{c}']. \end{align} In words: "zooming out" (i.e. increasing all distances by a constant factor) has the same effect as rescaling all coupling constants. If one gets all their signs and conventions right, they should find that as we zoom out, all but a finite couplings are irrelevant: they go to zero.
Note that in flat spacetime, we can afford to be sloppy and think about rescaling coordinates instead of the metric. In curved space, this doesn't work, and rescaling the metric (as described above) is the right way to "zoom out".
Quantum Gravity
In quantum gravity, we might try calculating amplitudes for the restriction $g_{\partial M}$ of the metric itself to the boundary $\partial M$. Ignoring matter, we could tentatively write: \begin{align} Z[g_{\partial M},\mathbf{c}]= \int_{g|_{\partial M}=g_{\partial M}} Dg \exp(-S[g_{\mu\nu};\mathbf{c}]). \end{align} We'd now like to ask the same question: how does the physics change as we zoom out? But now $g_{\mu\nu}$ is not a background field: we can't simply rescale it and ask how the dynamics changes. The whole previous discussion is not applicable.
Therefore it's not at all clear to me what it means to say things like "gravity is strongly coupled at high energies/weakly coupled at low energies". In fact, I don't even know what people mean when they say "gravity is nonrenormalisable". Does they just mean the perturbative statement? Or can the statement be understood nonperturbatively via a well-defined notion of RG flow in quantum gravity?
