Do "between" and "across" have distinct meanings in "between and across children in various sociodemographic groups"?

Question

Do "between children" and "across children" in following sentence have different meanings?

In addition, wide health inequalities persist between and across children in various sociodemographic groups in the United States. In spite of the changing nature of health care for children overall, child morbidity remains disproportionately high in children from lower sociodemographic homes. Children from more deprived and disadvantaged backgrounds tend to have poorer health and higher rates of mortality from major diseases such as heart disease, diabetes, and cancer in adulthood. Especially for children, the origins of poor health can be multiple and complex, and addressing the root causes can be equally complex. Wider determinants — the influencing factors responsible for everyone’s health— are more significant for children.

Friedman, H. S. (Ed.). (2011). The Oxford handbook of health psychology. Oxford University Press.

Answer 1

These prepositions highlight the difference between those children in the same and in different sociodemographic classes.

Between refers to those in the same class, and across refers to those in different classes. So it's saying that children in the same class don't have the same levels of health ("inequalities between...") and children in different classes also have different levels of health ("inequalities across...")