  We prove that the property of being cyclic subgroup separable, that is having
all cyclic subgroups closed in the profinite topology, is preserved under
forming graph products.
  Furthermore, we develop the tools to study the analogous question in the
pro-$p$ case. For a wide class of groups we show that the relevant cyclic
subgroups - which are called $p$-isolated - are closed in the pro-$p$ topology
of the graph product. In particular, we show that every $p$-isolated cyclic
subgroup of a right-angled Artin group is closed in the pro-$p$ topology, and
we fully characterise such subgroups.
