I'm actually a monarchist, but not like this. I just believe the current political system of Canada is strong and shouldn't change into a republic. I've never even heard of this movement. I hope I never have to again.
A few I can remember with constitutional monarchy: Australia, Belgium, Canada, Denmark, Japan, Liechtenstein, Luxembourg, Monaco, Netherlands, New Zealand, Norway, Spain, Sweden, and the United Kingdom.
What's more troubling is that there's very little history of constitutional monarchies reverting to absolutism and staying that way for a long time. On the other hand, many post-Soviet USSR republics (including Russia) provide plenty of democratic presidents turning into dictators.
Conventionally, though, these powers are used only through the advice of the Prime Minister, since the Prime Minister has some degree of democratic legitimacy (although interestingly enough, the Prime Minister is not elected directly by the people. The Prime Minister is Prime Minister only because he is the leader of the party that was chosen by the Crown to form government) and the Crown does not, since the Governor-General is appointed by the Monarch, with the advice of the Prime Minister, and the Monarch is obviously a hereditary position. Now, if the Crown were given democratic legitimacy, say through an elected head of state, suddenly the position is empowered and the stability of the system is called into question. Maybe the Crown decides not to sign a bill into law, or dismisses a minister, or dissolves Parliament for no reason, the Crown could say that he (or she) has the mandate of the people to do so, which is obviously bad for the country.
I'm not saying its necessarily bad for the Crown to be empowered like that, just that the system is stable and strong as it is, and it is not broken, so why fix it? It throws a lot of the political system into question and unbalances the delicate balance of our constitution.