Let $G$ be a simple, undirected graph with vertex set $V$. For $v \in V$ and $r \geq 1$, we denote by $B_{G,r}(v)$ the ball of radius $r$ and centre $v$. A set $C \subseteq V$ is said to be an $r$-identifying code in $G$ if the sets $B_{G,r}(v) \cap C$, $v \in V$ , are all nonempty and distinct. A graph $G$ admitting an $r$- identifying code is called $r$-twin-free, and in this case the size of a smallest $r$-identifying code in $G$ is denoted by $\gamma_r(G)$. We study the following structural problem: let $G$ be an $r$-twin-free graph, and $G^*$ be a graph obtained from $G$ by adding or deleting a vertex, or by adding or deleting an edge. If $G^*$ is still $r$-twin-free, we compare the behaviours of $\gamma_r(G)$ and $\gamma_r(G^*)$, establishing results on their possible differences and ratios.