Fertility rates have been decreasing rapidly all over the developing world. For instance, in Brazil it went from over 5 to around 1.7 in just 50 years. It has decreased so much that that country's very generous public pension system, created when the median age was around 18, eats most of the government's budget and has all but collapsed (current median age is 37).

Back in the 1970s your typical sci-fi dystopia was an overcrowded world. Personally I'm more afraid of a world with ghost cities inhabited by a handful of elderly people

The evidence is yes. Assuming the developing world also provides access to birth control, allows women to access education and jobs, etc. then it seems that cultures naturally reach a point where the birthrate lowers.

Based on what we know, the global population would crest, and then begin to gradually subside globally. The biggest question would be, what is the peak and is that number tolerable even for a short while.