Wikipedia has America free of dynasty but this is only by using an out dated definition under which the Carringtons don't count. A modern dynast is anyone for whom the pursuit of happiness revolves around conferring wealth and power to his offspring and his offspring's children by virtue of his own accomplishments.
To what degree does the American constitution protect us from dynasts?
First there is
SECTION 8.
The Congress shall have power to lay and collect taxes, duties, imposts and excises, to pay the debts and provide for the common defense and general welfare of the United States; but all duties, imposts and excises shall be uniform throughout the United States;
Clearly transfer taxes are within the powers of US government. But does the government have an obligation to prevent dynasty? There is the Fourteenth Amendment
Section 4. The validity of the public debt of the United States, authorized by law, including debts incurred for payment of pensions and bounties for services in suppressing insurrection or rebellion, shall not be questioned. But neither the United States nor any State shall assume or pay any debt or obligation incurred in aid of insurrection or rebellion against the United States, or any claim for the loss or emancipation of any slave; but all such debts, obligations and claims shall be held illegal and void.
All money is debt and when dynasty, as all dynasty must, comes around to challenging the government and enslaving its people then that debt is void. If we agree that dynasty inevitably leads to this challenge then the government has an obligation to use its legal taxation power long before its existence is threatened.
Does the government have a constitutional obligation to suppress dynasty even when its existence is not threatened?
There is Amendment 13
Neither slavery nor involuntary servitude, except as a punishment for crime whereof the party shall have been duly convicted, shall exist within the United States, or any place subject to their jurisdiction.
According to Wikipedia
Involuntary servitude is a United States legal and constitutional term for a person laboring against that person's will to benefit another, under some form of coercion other than the worker's financial needs.
A dynasty, is by nature, a form of coercion beyond financial needs. There is no legal way for me to agree to a form of labor that forces myself and all successive generations to an unequal division of profits based purely on lineage. Whether that coercion takes the form of violence or a company store debt slave based scheme is not legally relevant.