Hi
I have searched for a passenger list for the Carnatic for 1878, but have had no luck I am afraid.

I have found a newspaper article from the Lyttelton Times describing the voyage, which may interest you if you are satisfied the family came out on that ship. I can email it to you if you contact me off list. Nice quote:

"The whole of the immigrants look uncommonly well and hearty, and appear to be a really good and serviceable body of people." Note the emphasis on the qualities expected of immigrants to a colony! The writer might have said the same thing about an import of cattle or sheep!

Good luck finding a passenger list...
John