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?



Finite vertex-transitive graphs that look like infinite vertex-transitive graphs
For Cayley graphs, you're basically asking about residually finite groups.


A group $G$ is called *residually finite* if, for every non-trivial $g$, there exists a finite quotient $f:G\to Q$ such that $f(g)\neq 1$.


It's an easy argument that a finitely generated group $G$ is residually finite if and only if, for every $n$, there exists a finite quotient $f:G\to Q$ such that $f$ restricts to an injection on the ball of radius $n$ in the Cayley graph of $G$


In particular, the Cayley graph of any finitely generated group has your property if the group is residually finite (thanks to Boris Bukh for his comments below).


Many groups are residually finite---for instance, Mal'cev proved that all finitely generated linear groups have this property (brief proof---reduce modulo a suitable maximal ideal). 


However, there are non-residually finite groups. One of the simplest examples is the (2,3)-Baumslag--Solitar group with presentation


$BS(2,3)=\langle a,b\mid b^{-1}a^2b=a^3\rangle$ .


Of course, there are much more sophisticated examples---Higman constructed an infinite, finitely generated group with no non-trivial finite quotients, and there are even infinite, finitely generated, simple groups. 


