I'm not very familiar to LNC (only r26) but following the info from YG neither chaos thrust or moonrise have elemental damage. By their info, LNC really has no elemental WS.

Well, but maybe its because maybe they're holding it for elemental breathes since they're LNC/DRG "signature skills", just like jump. At FFTA series, the dragoons use the breath themselves, not requiring the pet as in XI so i'd expect similar feature here.