  In this paper we consider a family of non local functionals of
convolution-type depending on a small parameter $\varepsilon>0$ and
$\Gamma$-converging to local functionals defined on Sobolev spaces as
$\varepsilon\to 0$. We study the asymptotic behaviour of the functionals when
the order parameter is subject to Dirichlet conditions on a periodically
perforated domains, given by a periodic array of small balls of radius
$r_\delta$ centered on a $\delta$--periodic lattice, being $\delta > 0$ an
additional small parameter and $r_\delta=o(\delta)$. We highlight differences
and analogies with the local case, according to the interplay between the three
scales $\varepsilon$, $\delta$ and $r_\delta$. A fundamental tool in our
analysis turns out to be a non local variant of the classical
Gagliardo-Nirenberg-Sobolev inequality in Sobolev spaces which may be of
independent interest and useful for other applications.
