Any theory that extends general relativity must reduce to GR in some limit, and must (of course) agree with experiment. Probably the simplest such extension is Brans-Dicke gravity, which corresponds to replacing the gravitational constant by a scalar field on spacetime: $\frac1G\to\phi(x)$. The kinetic term for $\phi$ in the Lagrangian, appropriately dimensionalised, has a prefactor $\omega$. This field now couples to the Ricci scalar in the Lagrangian, and so has an equation of motion which is determined by the geometry and hence the energy-momentum content. Note that the only free parameter in the theory is $\omega$, which must be constrained by experimental bounds. Current observational evidence suggests $\omega>40\,000$ - for comparison, general relativity is heuristically reobtained in the $\omega\to\infty$ limit. Here's the equation of motion for $\phi(x)$:

$$ \square\phi=\frac{8\pi}{2\omega+3}T_\alpha^\alpha $$

which is a wave equation for massless $\phi$, sourced by the trace of the energy-momentum tensor. This means that $\phi$-waves propagate at the speed of light and, asymptotically, $\square\phi=O(\rho/\omega)$ for dust with matter density $\rho$. Now in the new "Einstein equations" relating $R_{\mu\nu}$ and $T_{\mu\nu}$, you can interpret $\phi$ and its variations either as a "generalised mass" or a "generalised spacetime curvature" - either way, in the Newtonian limit, it is manifest that this is equivalent to a changing gravitational coupling (even though the Einstein equations themselves have become rather more complicated).

With the reasonable additional assumption that $\phi$ approaches a constant value $\tilde\phi$ asymptotically, we can expand around this value in powers of $\frac{1}{\omega}$. This means that $\phi\sim\tilde\phi+O(1/\omega)$ (provided that the trace of the energy-momentum tensor does not vanish identically, see [1]). Since, as mentioned, $\omega>40\,000$, we would expect only very small variations in the gravitational strength. However, due to their long-range nature and luminal travel, it is plausible that many such waves, generated from various sources and different spacetime locations superpose to reduce $G$ in a region (bearing in mind that the paths of these fluctuations are themselves affected by the spacetime geometry, and so lens naturally into a dense region), or that there is some (possibly coincidental) fine-tuning. A good review is given in [2], albeit in asymptotically de-Sitter space (although a similar analysis applies).

References:

• [1] Banerjee, Sen (1997)
• [2] Ozër, Delice, Gravitational waves in Brans-Dicke Theory with a cosmological constant

Any theory that extends general relativity must reduce to GR in some limit, and must (of course) agree with experiment. Probably the simplest such extension is Brans-Dicke gravity, which corresponds to replacing the gravitational constant by a scalar field on spacetime: $\frac1G\to\phi(x)$. Leaving $\phi$ as non-dynamical leaves too many degrees of freedom that we need to put in by hand (the values of $\phi$ at every point in spacetime) - while this is a well-defined theory, it is not very predictive.

We remedy this by giving a kinetic term for $\phi$ in the Lagrangian, which when appropriately dimensionalised, has a constant prefactor $\omega$. This field now couples to the Ricci scalar in the Lagrangian, and so has an equation of motion which is determined by the geometry and hence the energy-momentum content. Note that the only free parameter in the theory is $\omega$, which must be constrained by experimental bounds. Current observational evidence suggests $\omega>40\,000$ - for comparison, general relativity is heuristically reobtained in the $\omega\to\infty$ limit.

In the new "Einstein equations" relating $R_{\mu\nu}$ and $T_{\mu\nu}$, you can interpret $\phi$ and its derivatives either as constituting a "generalised mass" or a "generalised spacetime curvature" - either way, in the Newtonian limit, it is manifest that this is equivalent to a changing gravitational coupling (even though the Einstein equations themselves have become rather more complicated). This means that regions where $\phi$ is large have a low gravitational strength and vice versa.

Here's the equation of motion for $\phi(x)$:

$$ \square\phi=\frac{8\pi}{2\omega+3}T_\alpha^\alpha $$

which is a wave equation for massless $\phi$, sourced by the trace of the energy-momentum tensor. This means that $\phi$-waves propagate at the speed of light away from regions where $T_\alpha^\alpha>0$.

With the reasonable additional assumption that $\phi$ approaches a constant value $\tilde\phi\sim\frac1G$ asymptotically, we can expand around this value in powers of $\frac{1}{\omega}$. Provided that the trace of the energy-momentum tensor does not vanish identically, this means that $\phi\sim\tilde\phi+O(1/\omega)$ and

$$ R_{\mu\nu}-\frac12 Rg_{\mu\nu}=\frac{8\pi}{\phi}T_{\mu\nu}+O(1/\omega) $$

If you imagine a single $\phi$-wave propagating through space, regions where $\phi(x)$ has a crest have below-average gravity while regions in a trough have above-average gravity. 

Since, as mentioned, $\omega>40\,000$, we would expect only very small variations in the gravitational strength. However, regions dominated by electromagnetic energy-momentum have $T_\alpha^\alpha=0$ and so allow $\phi$-waves to travel through them unimpeded, as in a vacuum (since the equation of motion for $\phi$ reduces to the free wave equation) - this is what is alluded to in their "long-range nature". So due to their long-range nature and luminal travel, it is possible that many such waves, generated from various sources and different spacetime locations superpose to reduce $G$ in a region (bearing in mind that the paths of these fluctuations are themselves affected by the spacetime geometry, and so lens naturally into a dense region).