Usually the flight resembles that of a Pintail, straight with rapid wing-beats, a resemblance accentuated by the slender neck and tail.Do you think the Luneau bird flies like a Pintail?
Personally, I think it flies a lot more like a Pileated. For one thing, the Luneau bird's wingtips are very high at the top of the wingstroke, like a Pileated but unlike a Pintail. (Of course, the exact wingtip positioning is easier to see on the videos if you step through them frame-by-frame, or if you "pause" them at various times).
Take a good look at these three videos and see for yourself:
Luneau bird
Pileated Woodpecker
Yellow-billed Pintail
The Pintails are not in powered flight throughout the entire video. However, the video shows a few flaps of powered flight as the birds fly to the left against a headwind, and it shows more powered flight as the birds turn to the right and fly away from the camera.
Note that I've tried to find some Northern Pintail video online, but have been unsuccessful so far. Please email me if you can help.