If so, how does this affect the historiography of China/East Asia? Did East Asian historians, like their western counterparts, cite the works of their predecessors from a previous era? Are there eras which are particularly under-represented? Was there a system of preservation similar to western monasteries c. 400-1400?
Every dynasty produced an official history of previous dynasties, and these are all extant. Furthermore numerous smaller histories remain along with court records. The debate over the salt and iron tax of Han Wudi, for example, is something of a minor classic of Chinese intellectual literature. The last major break in Chinese source transmission I can think of is with the fall of the Han, and for a thousand years thereafter the historical record is generally more complete in China than in Europe.
So I am not familiar with such a policy, nor any real policy procedure new dynasties followed at all, really. Each one came to power in rather different circumstances. Furthermore many emperor, such as Kangxi, were quite obsessive when it came to preserving literature.
It's possible that you slightly misheard a habit practiced by Qing dynasty emperors. Beginning with the Kangxi emperor, Qing rulers began to rely on a system of notes called palace memorials to receive reports, communicate with officials outside the capital, and govern the country directly and secretly, without any bureaucracy in between. The Kangxi emperor, and even moreso his successors the Yongzheng and Qianlong emperors used these messages extensively.
Officials and military officers throughout the country would send these memorials by special couriers, and only the Emperor would read them and personally write a response on the same page, sometimes just orders, sometimes a whole letter, sometimes just an acknowledgment that they received it, and the courier would bring it back. So, there were no copies, no secretaries involved, no middlemen learning state secrets. But— and this is the part you may have heard about?— when the Emperor died, and his successor took the throne, he ordered that all memorials written by his father be sent back to the Imperial palace, so that he could learn any secrets the previous emperor had thought of him, assert his authority by sending out his own commands that superseded all of his father's, etc. I don't believe the memorials were destroyed, as we have a lot of them, but they were completely removed from circulation.