  Let $X$ be a connected, locally finite graph with symmetric growth. We prove
that there is a vertex coloring $\phi\colon X\to\{0,1\}$ and some
$R\in\mathbb{N}$ such that every automorphism $f$ preserving $\phi$ is
$R$-close to the identity map; this can be seen as a coarse geometric version
of symmetry breaking. We also prove that the infinite motion conjecture is true
for graphs where at least one vertex stabilizer $S_x$ satisfies the following
condition: for every non-identity automorphism $f\in S_x$, there is a sequence
$x_n$ such that $\lim d(x_n,f(x_n))=\infty$.
