For a vertex-transitive graph $G$ and a positive integer $d$, and let $G(d)$ be the subgraph induced by all vertices of $G$ within distance $d$ of some given vertex $v$ (since $G$ is vertex-transitive, this doesn't depend on $v$).

Given an infinite (but locally-finite) graph $G$, does there always exist a finite vertex-transitive graph $G_d$ such that $G(d)$ is isomorphic to $G_d(d)$ (intuitively, a neighborhood of radius $d$ in the infinite graph "looks like" a neighborhood of radius $d$ in the finite graph)? If so, are there any good methods for constructing such a graph (given some nice representation of $G$, say as a Cayley graph of some infinite group). It seems that if $G$ is the Cayley graph of some group (which is true by Sabidussi's theorem), then it might be possible to take some presentation of this group and add some extra relations so it becomes finite, but I haven't been able to make this method work.

One example of a graph $G$ where such a family of graphs $G_d$ exists are lattice graphs $\mathbb{Z}^m$ (where points are connected if they are at unit distance); these are similar to toroidal grid graphs $(\mathbb{Z}/d\mathbb{Z})^m$. I can also construct other such graphs $G$ by "adding edges" to $\mathbb{Z}^m$ (by say, connecting all points at distance < 5). Barring a positive answer to the above question, are there any other graphs $G$ where such a family exists?