  For a graph $G$, two dominating sets $D$ and $D'$ in $G$, and a non-negative
integer $k$, the set $D$ is said to $k$-transform to $D'$ if there is a
sequence $D_0,\ldots,D_\ell$ of dominating sets in $G$ such that $D=D_0$,
$D'=D_\ell$, $|D_i|\leq k$ for every $i\in \{ 0,1,\ldots,\ell\}$, and $D_i$
arises from $D_{i-1}$ by adding or removing one vertex for every $i\in \{
1,\ldots,\ell\}$. We prove that there is some positive constant $c$ and there
are toroidal graphs $G$ of arbitrarily large order $n$, and two minimum
dominating sets $D$ and $D'$ in $G$ such that $D$ $k$-transforms to $D'$ only
if $k\geq \max\{ |D|,|D'|\}+c\sqrt{n}$. Conversely, for every hereditary class
${\cal G}$ that has balanced separators of order $n\mapsto n^\alpha$ for some
$\alpha<1$, we prove that there is some positive constant $C$ such that, if $G$
is a graph in ${\cal G}$ of order $n$, and $D$ and $D'$ are two dominating sets
in $G$, then $D$ $k$-transforms to $D'$ for $k=\max\{ |D|,|D'|\}+\lfloor
Cn^\alpha\rfloor$.
