Let $G$ be a graph, and for every $S \subseteq V$, let $N(S)$ denote the neighborhood of $S$. The vertex expansion of $G$ is $$ \min_{S\subseteq V, |S|\le |V|/2} \left\{ \frac{|N(S)|}{|S|} \right\}.$$

For the binary Hamming cube $\{0,1\}^n$, the vertex expansion is well understood: Harper's theorem says that the set of size $s$ of minimal-size neighborhood is the ball of size $s$.

Let us now consider the general Hamming graph: this is the graph whose vertices are the strings $\{0,\ldots,q-1\}^n$, and two strings are connected by an edge if and only if they differ on exactly one coordinate. My questions are:

Is anything known about the vertex expansion of the general Hamming graph? In particular, is there an analogue of Harper's theorem for this graph?

Harper's theorem actually says something stronger: Denote by $N_d(S)$ the $d$-neighborhood of $S$, i.e., the set of vertices of distance at most $d$ from $S$. Then, the set $S$ of size $s$ minimizing $|N_d(S)|$ is the ball of size $S$. Is there any lower bound about $|N_d(S)|$ as a function of $|S|$ in the Hamming graph?

Thanks!