Thank you for getting back to me. I hope that someone who recognises the family makes a connection with you and helps you to further your research. In the meantime here are some resources that may be of use to you:
Church Records:
Most Catholic records are held locally so you may need to write to the local parish priest for possible assistance. One website that you may find useful is the Irish Times where they give an overview of what records are available in specific parishes. It also shows you where copies of the records are available. For Magheralinn/Moira parish, follow this link:
http://www.irishtimes.com/ancestor/fuses/rcparishmaps/index.cfm?fuseaction=showidrecords&CityCounty=Down&parish=Magheralinn&churchid=438
For Belfast City/Ballymacarret:
http://www.irishtimes.com/ancestor/fuses/rcparishmaps/index.cfm?fuseaction=showidrecords&CityCounty=Down&parish=Ballymacarret&churchid=393
Most surviving Church of Ireland records are still held by the local clergy, although some are in the National Archives of Ireland and others are in the Representative Church Body Library in Dublin. Here are their websites: http://www.nationalarchives.ie/ and http://ireland.anglican.org/about/42
Lists of these surviving registers can also be found at the National Library of Ireland.
Land Record:
Have you tried looking for record of the family in Griffith?s Valuation? Thi is a property valuation survey that was carried out in Ireland 1848-1864. It can sometimes act as a good census substitution as there are no complete census records that have survived prior to 1901.
http://www.askaboutireland.ie/griffith-valuation/
Civil Records:
Civil registration began in Ireland in 1864 so any births, deaths or marriages in the family after this date would be recorded in these records. Civil records are available from the General Register?s Office. Here is their website:
http://www.groireland.ie/research.htm
You can search the indexes to these records up to 1958 online at:
https://familysearch.org/search/collection/1408347
Immigration Records:
Have you checked the immigration records in Australia? Generally, more information was given at the port of arrival rather than the port of departure. The University of Woolongong has produced, on microfiche, a complete index and transcript of all information concerning immigrants of Irish origin recorded on ships' passenger lists between 1848 and 1867 for New South Wales.
http://www-library.uow.edu.au/archives/collections/UOW095480.html
Some other websites that you may find useful are:
The National Archives of Ireland http://www.nationalarchives.ie/genealogy1/genealogy-records/introduction/
The National Library of Ireland http://www.nli.ie/en/family-history-introduction.aspx
The National Archives UK ? genealogy search: http://www.nationalarchives.gov.uk/records/looking-for-person/
The Public Records Office of Northern Ireland http://www.proni.gov.uk/index/family_history.htm
Family Search: www.familysearch.org
Genealogy Links: http://www.genealogylinks.net/uk/ireland/down/index.html
Irish Genealogy Toolkit: http://www.irish-genealogy-toolkit.com/index.html
Remember to post as much information as you can with regard to the people you are researching. The more information you post, the more likely it is that one of our volunteers will be able to advise or assist you. Also include information concerning which sources you may have already used so others may further your search.
Kind regards,
Genealogy Support