  Ryser proved that any two tournaments with the same score sequence are
$C_3$-equivalent while Beineke and Moon proved the $C_4$-equivalence for any
two bipartite tournaments with the same score lists. In this paper, we extend
these results to orientations of $G$ vertex-multiplications. We focus on two
main areas, namely orientations with the same score list and with score-list
parity. Our main tools are extensions of the refinement technique, directed
difference graph and a reduction lemma.
