There may be one trace left of Ellen in Ireland. Surviving records of the Parsonstown Workhouse show that two Ellen Dooleys were discharged from the workhouse in Parsonstown within a few weeks of each other in late 1849. This may well have been the future Mrs Murray and the future Mrs Flynn, but there is no way to link either girl with subsequent events in Australia.
There is no record of Mrs Murray entering the colony. When Eliza and Catherine arrived in July 1850, they told officials they had no relatives in the country, so we can assume the person who would marry John Murray arrived in the colony after that date. If she did, her disembarkation was not recorded in surviving lists of assisted migrants.
Government bureaucracy being what it is, this is strange. After 1850, assisted passages were paid from a fund administered by the colonial government. Residents in the colony paid the government passage money so their friends or relatives could join them in the new land. The government gave a ticket to the prospective immigrant. Rigorous record keeping was a necessity to avoid abuses of the system.
On the other hand, she might have bought her own ticket. In modern terms, that would be about as likely as someone on the minimum wage buying a first-class ticket on Qantas to London. It’s possible, but not likely.
There is plenty of documentary evidence as to what she did after she got married. She was pregnant, more or less continuously, until 1873. Eight children are recorded to the couple. Her husband died in 1903. She followed him in 1919.