|My ancestors were the few who stayed on board the Australia|
in spite of the disabled engine
For example, I had the passenger list for my ancestors, Giovantomaso Scioli and Lorenza Palladino, arrival in New York City on March 10, 1880, but frustratingly, the passenger list didn't mention the departure port or date. Wanting to know when and where their ship left from to get the full picture of their journey, I found NYC newspaper articles on March 10th and 11th, 1880, which told me not told me the ships departure details, but also that the ship made the journey on a disabled engine and hit a storm just before arrival!
That's probably a unique situation, but at the very least you should find a listing of ship arrivals with the details of its departure (example below). Sometimes, passenger names are even listed so if you're struggling to find any passenger list to begin with, you might want to try newspapers too (example below.