I wasn't sure how to make the title any more precise than that.

Let $(X,d)$ be a locally compact metric space. For any $\varepsilon>0$ let $\mathrm{Aut}_\varepsilon (X)$ be the subgroup of the automorphism group of $X$ generated by isometric automorphisms $f$ satisfying $d(x,f(x))<\varepsilon$ for every $x\in X$.

Suppose that for every $\varepsilon,\delta >0$ and any two $x,y\in X$ there exists an automorphism $f\in\mathrm{Aut}_\varepsilon (X)$ such that $d(f(x),y)<\delta$, does it follow that $X$ is bi-uniformly equivalent to $\mathbb{R}^n\times K$ for some finite $n$ and compact $K$? Where if $(Y,d_Y)$ and $(Z,d_Z)$ are metric spaces then $Y$ and $Z$ are bi-uniformly equivalent if there exists a uniformly continuous function $f:Y\rightarrow Z$ with uniformly continuous inverse.