See the Population Reference Bureau’s excellent video explaining the “Four Dividends” that countries generally attain following fertility decline as they pass through the demographic window. These four dividends are: (1) child survival, (2) educational attainment, (3) per-capita income, and (4) political stability (measured by 10-year risk of intra-state conflict).
Here are links to obtain the IUSSP Conference paper (authored by Elizabeth Madsen and me) that describes the timing of these changes, in terms of the movement of countries through the age-structural transition. A background paper on the Age-structural Theory of State Behavior is published in the Oxford Research Encyclopedia of Politics. Some of this information is published in a short essay on the “Eight Rules of Political Demography“, on the New Security Beat.