How is power transferred in states based on the three types of authority?
For this answer, I will be using the three-part typology of authority that Max Weber created. In each of these forms of authority, authority is passed down in a different way.
In a system that relies on charismatic authority, there is no set way for authority to be passed down. Authority depends on the person having the innate character traits that make other people want to follow them. Therefore, power cannot simply be passed from one person to another. If the first leader dies, another person must establish that they have sufficient charisma to attract followers. There is no official way to pass power down in such a system.
In a system of traditional authority, power is passed down based on unwritten rules. If, for example, the tradition says that power always passed from the paramount chief of a part of an island to his oldest nephew, then that is what will happen when the chief dies. There is a set way for power to be passed down, but it does not happen to be codified in written laws.
In a rational-legal system, the way for power to be passed down is codified in written form. For example, in the United States, the power of the president is passed from one president to whoever wins the next election. The rules for when the elections will be held and who may run in them are set in the Constitution.
Thus, each form of authority is passed down differently.