What evidence exists that the educational system causes the perpetuation of social inequalities? How and why does this happen?

In the United States, schools are funded by property taxes. This means that schools in wealthier neighborhoods are better funded than schools in poorer neighborhoods. Schools in wealthy, primarily white neighborhoods may offer AP courses and other forms of enrichment that increase students' chances of being admitted to college.
Next, there is what is called the "college wage premium." People who have graduated from college earn significantly more money than those who have not. Even more significant, students who attend highly selective colleges have higher lifetime incomes than those who attend less selective colleges. This reproduces economic inequality because only a small percentage of students from low-income families attend highly selective colleges.
Thus there is a cycle of students from wealthier families attending better funded primary and secondary schools, which serve as pipelines to more selective colleges, and then attending colleges whose graduates tend to earn higher lifetime incomes.
http://forum.mit.edu/articles/theres-a-paucity-of-high-ability-low-income-students-at-highly-selective-colleges-are-they-out-there/

https://www.insidehighered.com/news/2016/08/22/study-finds-graduates-most-selective-colleges-enjoy-earnings-payoff