Wish I could help with your Falveys. We have several of the same names, but my folks came to Rhode Island, USA about 1850.
I have Jeremiah Falvey married into my Sullivans family. Michael Falvey did as well.
I have a Humphrey Sullivan, which I thought was an unusual name.
I am quite sure they were from Kerry. The only naturalization I found was for Cornelius Ahern whose family were also connected and served as baptism and marriage witnesses. We visited Kerry last Sept. It is an amazing place. Good luck!