It is misleading to suggest that a free market means a level playing field or a pure meritocracy. Market economies do not exist in some ideal theoretical world with no history and no concrete realities.
People start from different places. If someone inherits several million dollars, and places it in safe investments, perhaps with the help of a financial advisor, that person needs no special talents to remain rich and pass their wealth down to their children. Even more, assortative mating makes it likely that wealthy people will marry other people from similar socio-economic backgrounds. Their children will be raised in a safe environment, attend good schools, have out-of-school tutoring, attend good universities, not need to work while going to school, and inherit wealth. They will have lived in areas and attended schools which enable them to develop networks and social connection with other wealthy people. On the opposite end of the spectrum, children from poor families lack many of the advantages of children from wealthy families.
Even worse, the growing disparity in the share of wealth stemming from capital over labor means that very few people can earn sufficient money to move into a higher economic stratum than their parents. This inequity especially affects black and Hispanic populations who historically have lower net worth than Caucasian ones.
In a pure market economy, there are no barriers to monopoly, rent-seeking, or structural inequality. This is why the United States has less opportunity for social mobility than European nations with a stronger social network and a more redistributive economic system.
Economic equity is very difficult to achieve in a free market economy because people naturally have different levels of ability and determination. In a free market economy, those differences end up leading to differences in economic status. Those differences can be passed down generation to generation until they become quite entrenched.
People may be born equal in the eyes of the law, but we are clearly not all born equal in every way. Some of us are clearly smarter than others. Some of us are more talented in particular ways. Some are taller, thinner, and/or better looking than others. Some of us seem to have personalities that cause them to be more driven and ambitious. All of these are attributes that can help people succeed economically in a free market system.
In a free market system, you can make more money as you have more of the attributes mentioned above. Free markets reward the talented and determined and punish those with lesser gifts. Over time, the gaps that this creates become more and more entrenched, when talented people marry other talented people and bring up children who may be more talented and who surely benefit from growing up in privileged circumstances.
It is hard (and perhaps impossible) to achieve economic equity (if, by that, we mean something like a situation where everyone has a roughly equal amount of wealth) because free markets naturally lend themselves to inequality.