This convergence was mostly driven by the fact that child-mortality rates declined the fastest among the poorest families.
In the countries where the gap increased, the study identified a common thread: poor governance.
The findings provide important information for making decisions about prioritising global health investments to effectively promote equity, said Eran Bendavid, assistant professor at the Stanford University's School of Medicine, and the study's author.
The study analysed data from nearly 1 million families living in 54 low- and middle-income countries to determine the relationship between mortality in children under the age of 5 and wealth inequality.
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"It's a pressing concern for many societies, especially in wealthy countries, but it's also been an issue in low- and middle-income countries," said Bendavid.
To compare wealth status and under-5 child mortality within a country, Bendavid used data from the demographic and health surveys for 1.2 million women living in 929,224 households in 54 developing countries.
Bendavid estimated the probability of a child dying before reaching age 5 per 1,000 live births.
To analyse trends in wealth status and under-5 mortality, Bendavid looked at all developing countries that had completed the surveys in two specific time frames: 2002-07 and 2008-12.
The average decline was 4.36 deaths each year per 1,000 live births among the poorest, 3.36 among the middle and 2.06 among the wealthiest. Because the poorest group's mortality rate is decreasing more quickly that the other groups, the gap in child-mortality rates is closing.
However, not all countries followed this same trend. In a quarter of the surveys examined by the study, inequality in under-5 mortality increased over time.
Bendavid found that four factors were present in countries with a narrowing child-mortality gap: government effectiveness, rule of law, control of corruption and regulatory quality.
The study appears in the journal Pediatrics.