I have a lead for a possible immigration record into the US, but I currently do not know if it connects with the family I'm researching. It does have a Patrick (Patk) Lambert traveling with a James (Jas) Lambert and what would be several siblings coming from Ireland in 1884 on the ship Spain. However, subsequent research as to whether this is the correct family remains to be seen. I've been trying to go through each child on that passenger list to see if I can make any connection, and it hasn't produced anything definitive. What's worse is that there's no 1890 census in the US, so a sixteen-year gap is a long time for life to happen.
The informant on the death certificate was John Lambert, Patrick's son.
Patrick did marry in Pennsylvania. The marriage record I see has a birth date, birth place, and residence at the time of marriage.