Labor has accused Malcolm Turnbull of having an unhealthy obsession with Kevin Rudd by getting into a schoolyard fight with the former prime minister.
Mr Turnbull lashed out at the former Labor leader, describing Mr Rudd's management of the people smuggling challenge as the worst policy failure in the nation's history.
The 2000 or so people in immigration detention on Manus Island and Nauru were there because Mr Rudd put them there.
"He let Australia down by abandoning a (border protection) policy that worked, in defiance of common sense and the results were tragic," the prime minister told reporters on the sidelines of UN meetings in New York.
Mr Rudd's response was to complain again about Mr Turnbull's decision to block his nomination as a candidate for the UN's top job.
Mr Turnbull has cited the lack of interpersonal skills and Mr Rudd's temperament as reasons for his decision.
Mr Rudd isn't buying the "frankly concocted excuse", saying it was at odds with Foreign Minister Julie Bishop's support.
And he dismissed speculation he could still clinch the gig if Hillary Clinton became US president as something from "la-la-land", saying he still required a nomination from Canberra.
"Good old Malcolm Turnbull decided one minute to midnight that was not to be the case. He didn't do it terribly elegantly," Mr Rudd told Sky News from New York on Thursday.
Labor leader Bill Shorten said the whole spat was a distraction from the important issues the prime minister should be focusing on during his visit to the US.
"I think that Mr Turnbull has an unhealthy obsession with Mr Rudd," he told reporters in Melbourne.
"Is that really all Malcolm Turnbull stands for? He goes to New York to have an argument with Kevin Rudd?"
Liberal senator James McGrath also weighed in from Down Under, tweeting: "Shut-up Kevin, just shut-up. No-one likes a whiner. #auspol #malcolmsomadetherightdecision"
Later, a spokesman for Mr Rudd said the former prime minister had no intention of responding to Mr Turnbull's "undignified, un-prime ministerial outburst" on asylum-seeker policy.
