I've lived in Africa, even lower middle class educated Kenyan's were on average are having 3-4 children.
At 13:50 in the video Hans Rosling does admit that Africa will continue growing for another 30 years, but will balance out the decrease in Asia's birth.
It's true, that culture and attitudes on getting more kids will need to change but providing education to women, better opportunities for men, and a stability in life etc. will make many want to have less kids.
Most countries in Asia have decreased the number of children as they prospered, even many muslim countries like Malaysia and Bangladesh. With the exception of the gulf arabs, as their income rose, the birth of children rose. But they aren't really educated tbh.