Sufficient conditions for quasi-isometric embeddings of Cayley graph

Question

I would like to know more about the assumptions under which the Cayley graph of a given group embeds quasi-isometrically into the space where the group is acting. For instance, if a group $G$ acts by isometries and properly discontinuously on a proper metric space $X$, with compact quotient $X/G$, then its Cayley graph quasi-isometrically embeds into $X$. I'm interested in this kind of results. Any comment would be much appreciated. Thank you.