If something is dynastic, it has to do with rulers or leaders who inherit their position of power. A dynastic business is run by successive generations of the same family.
If a country is ruled by a queen whose son will become king after she dies, that country is a dynastic monarchy. And if you know you'll become CEO of your family's trucking company, just like your father and grandmother before you, you can describe your position as dynastic. This word can be defined as "relating to a dynasty," and dynasty goes back to the Greek dynastes, "ruler," and dynasthai, "to have power."