Campbell is a very common name in both Scotland and Ireland. It’s going to be hard to trace the family without a little more information. When roughly did they move to Scotland? Did they have any children in Scotland, if so names and rough years of birth would help. What denomination was the family? Do you know whereabouts in Scotland they lived? When did they leave Scotland?
There’s a tree on Ancestry which has what appears to be this family, though I don’t see any immediate connection with New Zealand on it. In the 1861 census they were living in East Row, New Monkland. Francis was an Ironstone Miner. Family consisted of Mary 40, Francis 31, Mary 7, Peter 3, Francis under 1 year. Children all born in Edinburgh.
Peter Campbell reportedly died in Edinburgh 7.12.1929.
One tree has Francis dying in Castleblaney, Co Monaghan in May 1900. However I can’t find a death that matches that precisely. There was one in March 1900, registered in Castleblaney, but the man was a bachelor. So possibly that information about Francis' death is not correct.