When an empire falls and leaves no leadership, a power vacuum forms. This is when people fight for control of a specific area.
In Europe and China, after Rome and the Han fell, the administrative structures governing the empires and trade networks collapsed.
In Europe, this led to the rise of local, self-reliant, tribal governments and the early medieval feudal system. When the Han Dynasty fell in China, the empire was split into smaller kingdoms and even smaller independent states later.
Trade in China remained under threat from nomadic invasions by Mongols and Xiongnu. In Europe, trade was limited to local exchanges between manors.
Religion was a unifying factor in Europe and China at this time. In China, Buddhism became popular, while Christianity united communities in Europe.