I believe it is possible to promote just governance in developing countries without any cost to the developed country or at least minimal cost. I believe developed countries do have a moral obligation to promote this when possible. It is obviously the socially responsible thing to do and this world needs more social responsibility.
Yes, we are morally obligated to promote just governance in developing countries. It's kind of like being a big brother or a big sister - you look out for the youngest in the family. As fully developed countries we need to do our best to ensure that the people in developing countries are treated humanely.
Why should other countries have to police the world or promote a just country. If they become an issue then, bring in the troops. I think right now its everyman for themselves, it's not up to the world to justify the means for everyone else, they need to do it themselves.
Developed countries have no obligation, moral or otherwise, to promote anything in developed countries. The United States has done that for decades, it's called imperialism and interference. If we were truly interested in promoting just governance in developing countries, we would promote it in our own countries first. Something in the Bible about removing the plank in one's eye covers this, I think.