Kyiv ceased to be a great city in 1240 (the Mongols killed most of the population and burned down almost all houses), the most powerful Rus princes resided in other places as early as 1169, and the metropolitan archbishop of all Rus, despite leading a highly conservative institution, moved out of Kyiv in 1299.
If the Grand Duchy of Lithuania conquered Moscow and lesser Great Russian principalities by the late 14th century or later, Vilnius would have already been established as the capital of their empire, and they would have all the legitimacy they need from the simple fact of being victorious and benevolent rulers (the Gediminids (the ruling house of Lithuania) tried hard to respect ancestral rights of nobles in conquered lands, going as far as turning governorships into de-facto hereditary offices and allowing provincial assemblies to pass local laws even if they contradicted laws of Lithuania proper).
If they go Orthodox (which would be likely absent the union with Poland), it'd improve their legitimacy further, and the metropolitan would likely move his see to Vilnius as well (OTL, there were two archbishops, both recognized by the Patriarch in Constantinople, who claimed the title of 'Metropolitan of Kyiv and All Rus': the Muscovite metropolitan, with his de-facto see in Moscow, and the Lithuanian metropolitan, who alternated his residence between Kyiv and Navahrudak (a city in modern Belarus)).