1) My grandmother Kate Duffy's father was Michael Duffy... my grandmother mentioned he had come to America to work on the Brooklyn Bridge probably sometime in the the 1870's and sent money back to his family who lived on the lot of land 8 on the Woodfield historic maps. Is there a way to trace if in fact Michael did leave for America and return? Which record would most likely reflect those trips?
The only records I can think of are passenger lists and city directories, but the name Michael Duffy makes it difficult.
I searched the NY incoming passengers for 1820 to about 1892 at familysearch.org, looking for a Michael Duffy, born 1847 to 1857, arriving between 1870 and 1876.
There are 15. They didn't ask for contact information for the immigrants back then, like they did later when Kate Duffy and Michael Connolly arrived. You could go through the 15 of them and eliminate anyone traveling with a parent or wife, but that would probably leave more than one. I don't think you'll narrow him down any farther.
You can change the birth range if you think the one I used was too narrow or too wide.
https://www.rootschat.com/links/01sas/1875 New York City Directory - there are 5 men named Michael Duffy who are laborers.
1876 Brooklyn City Directory - there are 2 men named Michael Duffy who are laborers.
One of the lists had a Michael Duffy who was a [stone]mason. If that was part of the family story, it might be him.
There are UK Incoming Passenger Lists available on line starting in 1878. I didn't find Michael in that year.
A quick look for U.S. naturalizations did not find Michael.
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The Brooklyn Bridge, was built from 1869 to 1883, so the time frame is right on the dot. - You might ask 'how did he hear of the opportunity to work on the bridge?' and then see if there were advertisements in the Co Mayo papers, or if word was spread some other way. I'm pretty sure that sort of thing happened, but I don't know where I read it.
Along the same line, if you go through the passenger lists, see if you notice other single men with surnames common to the area Michael Duffy was from. One passenger list I saw had a Michael Duffy in with about 20 single men of the same age, and I wondered if they were all together in some way, or traveling for the same reason, e.g., building the bridge. But I'm not familiar with the surnames in Mayo, so I couldn't tell any more about them.