  This paper is concerned with qualitative properties of solutions to nonlocal
reaction-diffusion equations of the form$$ \int\_{\mathbb{R}^N\setminus K}
J(x-y)\,\big( u(y)-u(x) \big)\,\D y+f(u(x))=0, \quad x\in\R^N\setminus K,$$set
in a perforated open set $\mathbb{R}^N\setminus K$, where
$K\subset\mathbb{R}^N$ is a bounded compact "obstacle" and $f$ is a bistable
nonlinearity. When $K$ is convex, we prove some Liouville-type results for
solutions satisfying some asymptotic limiting conditions at infinity. We also
establish a robustness result, assuming slightly relaxed conditions on $K$.
