Read your post with some interest and allow me a couple of observations.
I'm assuming that with a location of Ulster and Greene Counties in NY State's Catskill region, that Michael Casey, whom you believe was a dish maker in Ireland, became a stonecutter in NY, presumably in the bluestone quarries of that area, an industry that along with the tanning industry brought the first significant waves of Irish immigrants into that area. Since Ulster County is in the Archdiocese of New York, you might already be aware that the baptismal, marriage and death records for the Archdiocese of NY are gradually being posted and available for free on the FInd My Past website. Greene County is in the Albany Diocese so sacramental events there might have to be researched at the individual parishes though there would not have been too many in that time period..
Also NYS death records are indexed beginning either in 1881 9r 1883 (working from memory here). Possible to check State censu records for 1875 and 1892 to see when Michael disappears from the censuses.
You may have documentation that you don't mention but it strikes me as a lot of movement for someone in the West of Ireland born in 1828 in Killarney and then going to work in Tuam but with a mother born in Tuam- how did she get to Killarney? Do you have her surname - perhaps an identifiably Kerry or Mayo surname? - how did you establish that mapping? The well-known Tuam is in County Galway rather than Mayo though the Diocese of Tuam may indeed extend into County Mayo (I wouldn't know that).
Happy hunting,
Ed Stewart