You're asking a very general question, so I'll answer more generally as well. If you need more specific regarding a specific region/time period in East Asia I can add more to supplement that.

You're confusing forms of a monarchical state (empire/kingdom) with the ruling family from which those monarchs come from, a line of hereditary rulers of a state (the dynasty). You can have an empire with a dynasty of rulers, and a kingdom with a dynasty of rulers. They are NOT mutually exclusive.

For example, the first long-lived empire in what is now China is considered to be the 漢 Han Dynasty, which lasted about four centuries. Its emperors were all from the 劉 Liu clan. Once the dynasty was usurped by a different clan (Wang 王 and Cao 曹, at different times), the empire's name was changed and the ruling house changed.

In China, at least, you almost never see a dynasty change without a corresponding change in the name of the state. But in Europe, that happened relatively often.