This is permitted by Standing Order 410(o), which gives as an example of breach of privilege
reflecting on the character or conduct of the House or of a member in the member’s capacity as a member of the House
I've long argued that this is rule is inconsistent with democracy and with freedom of speech in a free and democratic society. While the need to maintain order in the House means there's a need for such a rule in the debating chamber, in a democratic society there's absolutely no justification for such a rule outside of it. And it speaks volumes that its primary use thus far has been to punish a member of the public who criticised Peter Dunne's relationship with the liquor and tobacco industries. Now, its being used specificly to target members of the opposition for comments made outside the House. What are we, Nauru?
As for the process, Little and Hipkins will now be dragged before the Privileges Committee - a political kangaroo court with whipped votes. Its not a court, and there's no possibility of a fair hearing - whipped voting sees to that - and there's a high likelihood that they will be convicted on the (National) chair's casting vote. This is what passes for "justice" in our Parliament.
Of course, if the Speaker really felt that his character had been harmed by these allegations, he could sue for defamation. The fact that he and his party have instead chosen to pursue matters through the privileges committee tells us everything we need to know: he won't sue because he won't win. And he won't win because the fundamental allegation is true: he is biased, a patsy of his party. And no amount of show trials and forced, insincere apologies and extorted fines can disguise that.