Taking a stab at this even though I feel as though there are too many unknowns to answer it correctly. I could not access your link. I can suggest two different models which might get you started on answering the question and what kind of roadblock might stop or prevent--or cause troops to be called in.
First, what exactly does this "confederacy" control? If they are just spreading an ideology and have no real power, that's one thing, but if they are a trade group, that's another. If they "convert" a town which answers to a lord, does that mean the town no longer pays taxes or tribute to that lord? What happens to the people in power ABOVE the mayors and citizens they convert? In the 1500s, everyone paid out to everyone else. It is unrealistic to believe that no one would have noticed. If it in ANY WAY interferes with the flow of money into the pockets of those in power anywhere--troops will be called and there's no way it ends peacefully.
What do you consider "World-wide"? Exploration and other things will mean a culture clash for tribes that might not even think in the same terms as your confederacy. "global empires" that we've had have included Genghis Khan and Alexander the Great. And neither could hold it together for long. Rome is as close as we have come, so I suggest you take a look at their model--allow local custom insofar as it doesn't interfere with tax collection, and be there with military if the need arises.
If this doesn't involve money, AT ALL, in any way, and instead is about a philosophy and cooperation, consider using religion as your model. Christian religion is a good place to start. Notice that it hasn't taken over the world because competing religions and philosophies take hold in different places.
Your confederacy must change to fit the times, and be based on simple, easy to understand principles If an organization like this, which doesn't even really have nationalism going for it, is to survive for hundreds or even thousands of years to take over the whole planet, well...it's going to have to be adaptable and have very easy tenants which can be passed on.
Based on the spread of religion and cultures that I have seen here on earth, nothing actually EVER HAS taken over the whole planet in this way. The closest that I can think of is actually organized religion, but only if you're just talking Europe. It's pretty amazing that Christianity was all over Europe and the flippin' Pope at one point, had power over all the Kings because of the threat of excommunication. This happened by, oh 1300-1400, at the latest and, holy heck, that took what looks like about 1400 years from the crucifixion. Unfortunately, that's the KNOWN world, and that power has been waning.
Problematically you have to discover and conquer things before you can spread your culture effectively...