There was a dutch club in Christchurch - I think it might have been called Atlantis (after the ship they came over in ) and it may have changed it's name to a current club at some stage.
There were also heaps of Thistle clubs (Scottish) around - Auckland , Wellington ,Christchurch (?) Palmerston North , and Gisborne to name a few. There was a Chatham Cup Final in the 1930's I think between two Thistle sides.
There was also a club called NADA (Northumberland and Districts Association ) in Wellington in about the 1930's. Not sure whether that's ethnic ?
1934 was the Chatham Cup final between Thistle of Auckland & Thistle of Canterbury, the Auckland team won 2-1 in front of 7000 at the Basin. It was dubbed the "Scottish" Cup final. Wellington club Scottish Wanderers provided the hospitality including organising a procession with the teams to the Basin Reserve, led by a piper
Atlantis is now Cashmere Technical, thought it was english more than dutch, (but I'm not sure on that). formed in 1951 They changed their name in 1961 to Christchurch Wanderers, as they had adopted the colours of england's Wolverhampton Wanderers and there was no real link left with the Atlantis ship. Another name change in 1967, this time to reflect the area they were based, they became Cashmere Wanderers and then in 2012 merged with Woolston Technical to become Cashmere Technical
Island Bay United have always had a strong Italian immigrant connection as did Nelson United (now part of FC Nelson)
There were also Hungaria clubs in Christchurch (which was absorbed into the Atlantis club) and Palmerston North, but these didn't last long and never achieved the legendary status of Wellington's Hungaria