Well, in all three of those countries, Australia, New Zealand, and Ireland, this has happened.
In the 1990s, two Fianna Fáil prime ministers—taoiseachs—were removed from office because the supporting parties in the coalition, in one case, the Progressive Democrats, and in the other, the Labour Party, said they would not continue to support the party in power unless Fianna Fáil removed and replaced their leader, which they did. There had been some scandals that had arisen.
Now, as I mentioned quickly, in all of these cases the parliamentary party has the authority to remove the leader. That's not the case in Canada, unless you were to adopt the reform act. One of my concerns would be just what you suggest. Let's pretend that the Conservatives are in a coalition with the New Democrats, and the New Democrats demand that the Conservatives remove their leader in order to continue to have their support.