How many of them were booted from Britain to the penal colony? It may be that they have lost the details of their heritage, thus are left with just a general notion of their own history. My wife's paternal grandmother was a Cherokee indian. She lost her parents in the Trail of Tears march. That line of research is pretty limited.
My paternal grandmother was a very diligent researcher. Her side of the family arrived on the Mayflower. My paternal grandfather traces his family through his father's arrival in 1865 from Wales. My maternal grandfather traces his roots back to Jamestown. John Alden and Priscilla Mullins are in his direct lineage. At one point his family held title to what is now Norfolk. The grant from the King titled the area 'Fanshaw's bottom'. There are still streets with the family name in Norfolk.
They were actually pretty much voluntary migrants from the lower to mainstream middle class in Victorian Britain as NZ was never a penal colony. But from what I found at museums and government archives, it seems most migrants were either the Cornwall area, Oxfordshire, or northern England's industrial towns, and with some loewland Scots added into the mix.