Let's see if I have the right idea and if I can add a summary of what is known so far. You are seeking to track the children of ...
James McCarthy b in Ireland immigrated to USA abt 1850 (per patrice, no source given) married Ellen NN in 1864; she immigrated 1864 ( " )
You mention the death dates and places for James and Ellen but I don't recall sources being mentioned...? A family oral history is a legitimate source. Any source will do, as long as it's not just you saying the fact.
CHILDREN * Martin Mark born 28 July 1865 NY died Omaha NE 10 Mar 1905 " married Julia Corrigan (father Thomas Corrigan in Sand Creek, Holt, NE) * Michael McCarthy b abt 1866 NY * Julia McCarthy b abt 1868 NY * James McCarthy b abt 1870 NY * Peter McCarthy b abt 1873 NY * Patrick McCarthy b abt 1877 NY * John R McCarthy b abt 1882 KS (per patrice; no source given) * Ellen McCarthy b abt 1883 KS (per patrice; no source)
1870 census indexed as Carty or Casty
1870 Town of Phillips, Orleans, New York James Carty 46 b. in Ireland, farmer Ellen Carty 24 b Ireland Markie Carty 5 b NY Michael Carty 4 b NY Jul*ie Carty 2 b NY James Carty 4 months old b NY
1875 Geneva, Ontario, New York James Mccarthy 42 b 1833 Ellen Mccarthy 33 b 1842 Martin Mccarthy 9 Michael Mccarthy 8 Julian Mccarthy 6, daughter James Mccarthy 4 Peter Mccarthy 2
1880 Plainville, Rooks, Kansas James McCarthy 54 b Ireland, Farmer, parents b Ireland Ellin McCarthy 38 b Ireland, Keeping House, parents b Ireland Martin McCarthy 14 b NY, Farm Hand, parents b Ireland Michal McCarthy 13 b NY, Farm Hand James McCarthy 11 b NY, Farm Hand Peter McCarthy 7 b NY Patrick McCarthy 3 b NY
A candidate for Peter
1910 Eagle, Colorado Peter J McCarthy 37 b abt 1873 New York Tillie McCarthy 37 Tillie Darling 32
a Michael McCarthy, died 1930, is buried in the same cemetery as this Peter (in Denver, Colorado).
Do you have the death record for your ancestor Martin Mark McCarthy? Or his birth or marriage record? Or an obituary? Is his mother's maiden name listed in any of these?
You mention that you believe James became a flagman for the railroad - source? Can you say more about that?