I can think of two counter-examples:

* Polio eradication

* The Montreal Protocol on Substances that Deplete the Ozone Layer/Vienna Convention for the Protection of the Ozone Layer

Both are examples of multi-state cooperation that dealt with serious world-spanning issues, the Polio eradication has had the most immediate impact, but Ozone Layer depletion has been checked and the numbers will trend down to 1980s levels over the next 20 years.