Final answer:
The "lustrous" band of the Mongol Empire destroyed in 1502 by the Crimean Khanate was the Golden Horde. It had its origins as a part of the larger Mongol Empire but asserted independence and controlled substantial territory before fragmenting and eventually being absorbed by the Crimean Khanate.
Step-by-step explanation:
In 1502, the last surviving remnant of the "lustrous" band of the Mongol Empire was destroyed by the Crimean Khanate. This "lustrous" band is referred to as the Golden Horde. The Golden Horde was a Mongol and later Turkicized khanate established in the 13th century. It originated as a part of the Mongol Empire before gradually drifting to independence, controlling parts of Eastern Europe, the Caucasus, and Central Asia. Despite the fall of its capital, Sarai, in 1556, the Golden Horde persisted in various forms until its final dissolution.
Russian princes went to Sarai to pledge allegiance to the Mongols and were granted a jarlig, which confirmed their authority from the Great Khan himself. Although the Golden Horde asserted independence in the late 13th century, it faced continual external and internal conflicts. After being defeated by a united Russian army in 1480, the Horde fragmented into smaller units that merged with other Russian city-states. The most notable successor state was the Crimean Khanate, which finally annexed the Golden Horde territory in 1502 and was in turn annexed by Russia in 1783.
The territories ruled by the Golden Horde, which included parts of northwest Asia and the lands of the Rus, were notable for being less economically developed and having more diverse lifestyles than other parts of the Mongol Empire. This region avoided civil war for a longer period after the fall of the Il-Khanate, thanks to its less centralized authority and more independent structure.