Historically, why did Japan and Russia move their capitals, and what were the effects?
Peter originally moved the capital from Moscow to Petrograd because a) it was closer to the West, which he sought to emulate and maintain close contact with, and b) to assert his authority in the face of the nobles and priests who had a firm grip on Moscow. The capital was changed back again at the end of World War I and remained Moscow because it's far away from any frontlines.
As for Japan, IIRC in the eight century it changed from Nara to Kyoto because the imperial court wanted to get away from the Buddhist priest class which had grown too powerful. While Kyoto remained the seat of the emperor until the late 1800s, a parallel pseudo-government (the shogunate) evolved from the 12th century onward with its seat in a different city, first Kamakura, then after 1600 Edo/Tokyo. This shogunate slowly but steadily gained more and more power as the emperor lost his, until the emperor was only in charge de jure, while the real power lay with the shogun. With the Meiji Restauration in 1868 the shogunate was abolished and the emperor moved to Tokyo to make use of the already existing administrative infrastructure there.
All of these events can easily be translated into a one time stability boost upon completion of a Palace and/or free civics changes if we are talking game mechanics:
Moscow->Petrograd: Adoption of Centralism and Regulated Trade
Petrograd->Moscow: Adoption of State Party, Ideology, Totalitarianism, Central Planning etc.
Nara->Kyoto: Clergy->Monasticism (what followed this move was an attempt to get Buddhism to keep from interfering in government policy and focus purely on spiritual stuff)
Kyoto->Edo: If we count the Edo Shogunate (1600) as this Adoption of Isolationism and Caste System, if Meiji Restoration (1868) then Adoption of Colonialism/Nationhood/Conquest and Individualism and Constitution and maybe some other stuff.
The hardest thing to represent would be Peter's move away from Moscow because in the game the only tangible benefits would be a little bit more help in cultural pressure and maaaaaaaaybeeeeee a somewhat better capital for commerce which would benefit from Regulated Trade, which is probably never worth the now increased city maintenance costs along with the actual hammer cost for the Palace itself.