In what ways did the civilizations and empires of ancient India and China lay long lasting social and political foundations

They both start around the year 2000 BCE, four millennia ago, a bit later than Egypt and Sumer (Babylonia).

The first empires in India started around the Vedic period (1500-800 BCE). The Vedas are some of the oldest mythological texts in history. They laid the principles of Hinduism (which is one of the major religions even nowadays) and traditional Hindu philosophy. Since then, a series of different empires have ruled over the Indian subcontinent for more than three thousand years, among great wars and shifts in religion. Some examples are the Mauryan empire (321-185 BCE) or the Chola empire (850-1279 CE).

The Xia dynasty (2070-1600 BCE) was the first in ancient China. Since then, great emperors have ruled over the whole territory, but the country has also split apart into feudal kingdoms a lot of times, and subjected by foreign invaders like the Mongols. Some of the most important dynasties were the Tang Dynasty (618-907 CE) which flourished due to commerce, the Ming Dynasty (1368-1644 CE) a brilliant period for the arts, or the Qing Dynasty (1644-1912 CE) which lasted until the arrival of communism and gives us an idea of the extraordinary continuity of Chinese tradition and political system. Confucianism and Taoism were ancient moral and religious teachings that shaped Chinese society. Both China and India were strongly influenced by Buddhism.

(Note: Dates by the Encyclopedia Britannica)