Blacks in (increasingly, non-PC) Australian means [native] aborigines. Robyn is saying that the natives were dumped = (hastily / carelessly) moved / relocated to (charitable, church-sponsored) missions by (white) people who thought they were "just doing their job" (looking after and bringing civilisation to those natives).
As most people would now agree, that policy didn't work out too well. But to the average Australian today, that cynical reference to a failed historical policy doesn't exactly "come out of nowhere". For many, it's something they're very consciously aware of in any context involving aboriginals (it's not exactly the Holocaust, but as a nation, Australia isn't too comfortable with what they did).
I haven't seen the movie, but a quick glance at the subtitles implies the immediately-preceding context involves aboriginals. So the cynical reference to that particular example of the problems caused by people "just doing their jobs" would be completely natural and contextually relevant.