The journey of Niger is especially interesting. Between 1947 and 1983, the birth rate actually increased despite infant mortality declining. Do you happen to know why that is?
I would (cautiously) speculate that when an oppressive force (either political or natural) is abruptly removed from a population, that population will tend to experience increased birthrate and decreased child mortality simultaneously. Extreme poverty is such a hardship that it can lower the birth rate (in addition to its obviously adverse effect on child mortality). When the force is removed (e.g. by a regime change, or a period of economic growth), child mortality decreases due to better health and better food security. Simultaneously, people have access to a luxury that they may not have before: the opportunity to start families. So the birth rate can rise as well. It takes another generation before the positive effects (better health, stable food supply, etc.) to translate into better education, higher economic mobility, and ultimately lower birth rate. So perhaps independence from a colonial power that often used military force against the civilian population is the cause of this simultaneous increase in child survival and birth rate, which we see in several of the poorest African nations shortly after their independence.
As for the charts, they're among my favorite data visualizations. They were invented by Professor Hans Rosling, who you may know from this very animated TED talk [0]. Rosling passed away last year, but his book Factfulness was just released (co-authored with his children). It's all about the preconceptions and biases that people in the "developed world" carry with them that prevent them from thinking rationally about poverty and global development. The book is well worth your time.
[0]: https://www.ted.com/talks/hans_rosling_shows_the_best_stats_...
Looking at other countries I think the effect is strongest under 120 infant morality and there are plenty of short term counter examples like the US from WWII to 1958. Or Albania from 1932 to 1968.