Stephen,
The events are before the start of statutory birth, marriage and death registration in Ireland. To trace the family you would need to rely on church records. You haven’t said what denomination they were but I would guess Presbyterian both by the name and the fact that the majority of the population of Larne were of that denomination. The main Presbyterian church there was the non Subscribing church. It has baptisms for 1720-69, 1798, 1801 – 1803 & 1826 – 1829. Marriages cover 1721 – 1769 & 1826 – 1908.
There is a second Presbyterian church in the town but it only has marriages from 1846 and baptisms from 1861. The RC church’s records start in 1821 and the Church of Ireland in 1806.
The RC records are on-line on the nli site, for the others you would need to get someone to go to PRONI (the public record office) in Belfast and look them up.
Regarding passengers details from Ireland to NB, I would suggest you search on Ancestry or findmypast (subscription sites) as they have most of the records. However if the family left Ireland/Britain for Canada before around 1850, then there are few passengers lists, as they weren’t required then.
Though most Irish censuses pre 1901 have been destroyed, part of the 1851 census does still exist and it includes Larne and some of the surrounding area. I don’t see this family in it, and so suspect either they had left by 1851 or lived outside the area covered by the census fragments.
http://www.census.nationalarchives.ie/search/
