Is it true that the descendants of the Byzantine emperor served Russia?

The Russian princely family of the Cantacuzenes traced their descent from the Byzantine Emperor John VI Cantacuzene, who reigned in the mid-14th century. At least, that's what their family history claims. However, there is no documentary evidence of this connection.

After the fall of Constantinople, the Cantacuzenes continued to live in the city, which became Istanbul, and then Wallachia (future Romania), where they held high government positions. In the 18th century, part of the family moved to Russia, where they primarily distinguished themselves in military affairs. 

Some earned the right to command positions in wars against the Turks, the French and the Germans. Then, Major General Nikolai Cantacuzene became the appointed ataman of the Buh Cossacks.

Nikolai Cantacuzene.

Gregory Cantacuzene, meanwhile, distinguished himself in diplomatic service. He worked in various European countries and in the United States, where he contributed greatly to the proper representation of the Russian Empire at the 1893 Chicago ‘World's Fair’ – one of the largest in history.

After the 1917 Bolshevik Revolution, all the descendants of the Cantacuzenes left Russia and scattered around the world like most other opponents of Soviet power.