  We study graph $C^*$-algebras equipped with generalised gauge actions, and
characterise in terms of groupoids and groupoid cocycles when two graph
$C^*$-algebras are isomorphic by a diagonal-preserving isomorphism that
intertwines the generalised gauge actions. We apply this characterisation to
show that two Cuntz-Krieger algebras are isomorphic by a diagonal-preserving
isomorphism that intertwines the gauge actions if and only if the corresponding
one-sided subshifts are eventually conjugate, and that the stabilisation of two
Cuntz-Krieger algebras are isomorphic by a diagonal-preserving isomorphism that
intertwines the gauge actions if and only if the corresponding two-sided
subshifts are conjugate.
