...Earnings mobility for children from the very broad middle—parents whose income ranges from the bottom 10 percent all the way to the cusp of the top 10 percent—is not tied strongly to family income. These children tend to move up or down the income distribution without regard to their starting point in life. This may be one element of insecurity among the middle class: in spite of their best efforts, their children may be as likely to lose ground and fall in the income distribution as they are to rise.
The situation is very different for children raised by top-earning parents...
Much more here.