A pilot project available to search at familysearch.com. This project offers a digitised version of the Index to Bounty Immigrants, 1828-42 which was compiled by the Church of Latter Day Saints.
https://trove.nla.gov.au/newspaper/?q=This Guide highlights the key records and available indexes, relating to passengers arriving in New South Wales, 1788-1922. While most of the records relate to passengers disembarking in Sydney, records of arrivals in other ports are also included. Records relating to departures are also listed.