The C17 Globemaster III is a large (but not the largest) United States Air Force transport plane, used for strategic airlift, troop transport, cargo transport, tactical airlift, medical evacuation, and airdrop across the world.
It performed a brief display before disappearing off to the south. The commentator seemed surprised by this, so maybe something didn't go completely according to plan. Total loaded weight is 265 tonnes, which is almost 4 times more than the Hercules, which seems big enough on the ground!
This movie shows the initial pass from right to left. The C17 first flew in 1991 and is used by the US and several overseas air forces. It has 4 Pratt & Whitney F117-PW-100 turbofans, producing 40,440 lbf (180 kN) of thrust each, and a fuel capacity of 134000 L.
This is a second movie, going in the opposite direction. I apologise for the break half way, where the aircraft flew directly in front of the Sun. That great weather in Wanaka does have some negative effects!
Sources: Warbirds Over Wanaka 2024 program, Wikipedia.