As far as I can tell the only comprehensive information about migrants to Australia was gathered on arrival in Oz. Shipping agents in the UK and Ireland rarely kept records (or none that have survived). There must have been hundreds of agents in business judging by the number advertising in the papers. I have a random page from the Liverpool Daily Post for 29th April 1863 (which I can send you if you want) and most of the page is devoted to adverts for ships to Oz, New York, New Brunswick, Glasgow, Le Havre, Boston, Auckland. You name it!
In addition to adverts in the newspapers, most towns in Britain and Ireland had shipping agents who were often folk like the Postmaster or a Hotel owner, who got a little commission for taking bookings. Sort of early travel agents.