Let $\mu$ be a Borel measure on $\mathbb{R}^d$ ($d>2$) so that $$\int_{\mathbb{R}^d} \frac{d\mu(x)}{1+|x|^{d-2}}<+\infty$$ The potential $$\Phi(x):=\int_{\mathbb{R}^d} \frac{d\mu(y)}{|x-y|^{d-2}}$$ is then well-defined, a.e. $<+\infty$ and superharmonic. The 'average' of $\Phi$ over the sphere $\partial B(0,R)$ goes to zero as $R \to +\infty$ showing that $\Phi$ is 'grounded' more-or-less at infinity. My question is whether it is possible to construct, for arbitrary $x\in \mathbb{R}^d$ and arbitrary $\epsilon>0$, a continuous or $C^1$ path $\gamma_{x,\epsilon}:[0,1] \to \mathbb{R}^d$ starting from $x(=\gamma_{x,\epsilon}(0))$, which "descends along $\Phi$", i.e. $\Phi\circ \gamma_{x,\epsilon}$ is non-increasing, and so that moreover $\Phi(\gamma_{x,\epsilon}(1))\leq\epsilon$.
My own attempts/insights:
If we assume that $d\mu=\rho d\lambda$ with $\rho$ a continuous function, $\Phi$ is continuously differentiable and we can construct paths using a gradient flow formalism. That gradient flow is measure-increasing which might be used to show that for a.e. initial $x$ (see terminology above), that gradient flow does not terminate. But how to show that it descends for a.e. $x$ to a potential below $\epsilon$?
If I assume that $\Phi$ goes to zero at infinity, or otherwise that $\Phi^{-1}([0,v))$ is connected, and if moreover assume that $\Phi$ is $C^{d-1}$ (this requires the $\rho$ from previous point to be locally $C^{d-3,\alpha}$?) I think I can use Sard's theorem to "string together" a (piecewise $C^1$) path which descends in $\Phi$ most of the time, possibly interrupted by a countable number of arbitrarily short (in terms of the Euclidean metric on $\mathbb{R}^d$) intervals where it ascends.
Possibly my question contains a ton of "unnecessary superstructure" (e.g. the superharmonicity of $\Phi$ perhaps only plays a role in that it prevents that $\Phi$ has local minima).