This video nicely shows a kereru/NZ pigeon display flight. Check it out and you'll see how distinctive these displays are.
https://youtu.be/cSIUfncPzaU

The Great Kererū Count is on now, helping to gather information on the abundance and distribution of the New Zealand pigeon — also known as the kererū, kūkū or kūkupa. If you have any observations to add to the Display Flight Project between 19 and 27th September, why not add them to the GKC as well!
https://inaturalist.nz/projects/great-kereru-count-2015