You can make either the raft or the barge fly using the grappling hook. If you build a structure on the barge (as most people do) and grapple to the barge itself (not any foundations or structures on the barge) you are able to move the barge by reeling in. Because the barges (and big rafts) can be moved by the grappling hook this causes the barge to move towards your direction as you reel in. As you are above the barge in this scenario it moves up towards you, as long as you keep reeling it "flies" higher. Walking around the grapple point and releasing reel to drop a little allows the barge to be effectively steered or "flown" through the air.

As a few tribes on a private server I help run have barge bases I would like to keep the mod, however other tribes have started "flying" rafts to the volcano for metal and obsidian and its a little absurd.

The vanilla rafts do not have this issue as you reel towards the raft (the grapple can't move them).