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I confirmed everything you said...1860, 1870, and 1880 census records. The eldest child, Mary, is consistently listed as having been born in 1854, in New Jersey. I could not locate them in the 1850 census, however. Maybe they arrived just before Mary's birth? I also checked for them in New York in 1850. The Civil War records for Michael are fascinating...the NJ 6th infantry regiment fought in 18 battles, including the biggest ones. He was in for the whole thing. This is a surmise, but my husband's family is from Cregduff, near Ballinrobe in County Mayo, Ireland, and we think yours might be, too. We found them by first doing a surname concentration search for Duddys, and found that there are essentially two groups in the 19th c: the one in Mayo and one in northern Ireland. The recurring first names in the two groups are different, however, and ours match yours.