I can confirm that James (sibling to Catherine and Matthew) also immigrated to Australia. James arrived in 1875 with his young family Mary (nee Hacket, married in Knockavilla, Tipperary in 1863, with a record reference to living in Solohead at the time of marriage) and their 4 children - Thomas (born in 1864, with a birth record listing Kilshinane as place of birth and residence of family), Edmond (born in 1865 in Kilshinane), Bridget (born in 1866) and Honora (born in 1872, also recorded as Hannah and as Nora). All 6 arrived at Rockhampton, Queensland, Australia on 1st August 1875 on the Lady Douglas Ship.
James died in 1880 in Rockhampton, Queensland.
Bridget died in 1892 in Rockhampton, Queensland.
Edmond died in 1911 in Rockhampton, Queensland.
Honora (listed as Nora on death record) died in 1914 in Rockhampton, Queensland.
Mary died in 1918 in Rockhampton, Queensland.
Thomas died in 1933 in Capella, Queensland (buried in Clermont, Queensland).
Of the 4 children by James and Mary, Thomas (1864-1933) was the only one to be recorded as married and have children of his own. Death records for the other 3 children of James and Mary list them as single with no known children.
Thomas Dwyer 1864-1933 (eldest son to James Dwyer and Mary Hackett, grandson to Thomas Dwyer and Bridget Cleary) married Bridget Cecilia Walsh (1882-1967) in 1901 in Capella, Queensland and they went on to have 13 children (all living through to late adulthood.
James Dwyer - Find A Grave Memorial ID 160408681
At the time of posting this, his Find A Grave incorrectly lists his name as Thomas James Dwyer. James' Tipperary Birth record only lists his name as James, not Thomas James, his marriage and death records list him as James.
The information above has been provided by one of direct descendants of James (1838-1880) and Thomas (1864-1933), and confirmed with DNA matches to the descendants of James's brother, Matthew Dwyer (1825-1867) who immigrated to Victoria with Catherine Dwyer (1831-?). It also appears that their brother Timothy Dwyer (1822-?) immigrated on the same ship with Catherine and Matthew.