Such a move in Canada would backfire electorally. People here think the PM is like the President, so trying to oust him would be like a coup, and would sink the party deep in the polls for doing something undemocratic. A party would have to be absolutely toxic to risk that.
It's more that such processes don't have formal mechanism it seems. When Gordon Campbell quit it was because the caucus was moving against him, but not because someone called a formal vote. It was simply and painfully made apparent to him that he did not have the confidence of the party.
The closes similarity I can think of in recent years was the fall of Carol James' leadership of the BCNDP. That was fairly swift, and ended in a caucus meeting in Vancouver where the handing out of pro-james scarves was somewhat polarizing
