Consider a $d$-regular infinite transitive expander graph $G$, and let $B_r$ be a ball of radius $r$ in $G$. Can one place any upper bounds on the expansion of $B_r$?

My intuition is that $B_r$ will have very low expansion, perhaps exponentially small in $r$. The reason for this is that expander graphs are locally tree-like, and finite trees -- which is what I am imagining obtaining upon restricting to $B_r$ -- are the worst possible expanders.

I am thinking in particular about the case where $G$ is the Cayley graph of an infinite discrete group with exponential growth, but would be interested in understanding the answer more generally.