Hi Therese,
Thank you for your message.
Have you tried looking in to Patrick and Thomas's transportation records? The records of the Chief Secretary's Office, which had responsibility for the Penal system, are the major Irish source of information on transportees. Not all of the relevant records have survived, particularly for the period before 1836, but what does exist can provide a wealth of information. The records were formerly housed in the State Paper Office in Dublin Castle, which is now part of the National Archives of Ireland (NAI). These records were microfilmed and a database was presented to the Australian government and can now be found in many State archives. The NAI retains copies and the database, in particular, can save a great deal of time and effort. www.nationalarchives.ie
If you can find their records you may be able to find where they lived in Ireland, or at least where they were convicted and tried.
You can also check the Tithe Applotment Books. These are land records that compiled between 1823 and 1837 in order to determine the amount which occupiers of agricultural holdings should pay in tithes to the Church of Ireland. I did a search for Michael Mullavey in Co. Cavan, and one result came back from the Templeport parish in 1826. There is a good chance that this is your Michael. You can see the results here:
http://titheapplotmentbooks.nationalarchives.ie/search/tab/results.jsp?surname=mullavey&firstname=Michael&county=Cavan&parish=&townland=&search=Search
This is a good starting point for your research. You can use this parish name to search in Church records also for more information.
You can use the irishtimes website to see what Catholic records are available in the parish. Here is the link:
http://www.irishtimes.com/ancestor/fuses/rcparishmaps/index.cfm?fuseaction=showidrecords&CityCounty=Cavan&parish=Templeport&churchid=852
For Church of Ireland records check http://ireland.anglican.org/about/42 and http://www.nationalarchives.ie/.
Some other websites which may help you in your research 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: https://familysearch.org/
Genealogy Links: http://www.genealogylinks.net/uk/ireland/cavan/index.html
I hope that this helps.
Kind regards,
Genealogy Support