A Modern Example:
Numerous Ukrainians survivors have had their homes destroyed by Russian missiles. If the Russian prevail in this conflict, then I would think that said survivors can not claim damages against the victor. On the other hand for the purposes of the question, let us assume that the Ukrainians prevail in this conflict.
How & in what forum would a Ukrainian survivor claim / recover damages from Russia for the destruction of his / her home?
The Russian state is sovereign. This means the Russian government has to tell the Russian courts that they allow people to sue Russia for specific, enumerated things. If the Kremlin has not done so, then Russia has total, sovereign immunity from such claims.
In the united-states, the FTCA regulates how and when you can sue the USA or federal institutions for torts, while FSIA regulates how or when you can sue other countries in the USA.
In russia, the Federal Law No. 297-FZ of November 3, 2015 "On Jurisdictional Immunities of a Foreign State and Property of a Foreign State in the Russian Federation" seems to be a very similar law to FSIA. I could not determine if an equivalent to the FTCA exists.
Besides a law that allows claims against Russia, a Ukrainian-Russian peace treaty might contain a passus about reparations.
There are three very traditional ways how such handle reparations: In one case, such a treaty could establish that Russia pays the reparations to Ukraine, and then claims against Russia are paid out by Ukraine. In the other case, the treaty provides a framework to sue Russia for compensation. And in the last variant, no compensation for civilians is agreed upon at all.
Assuming your condition that Ukrainians prevail in this conflict, some kind of peace agreement will be signed. This agreement may include that is called reparations - the compensation for the war damage paid by a defeated state.
To be complete with the answer, there are debates ongoing about the theoretical possibility for seizing more than $300 billion in Russian central bank assets stashed in Western nations. There is currently no consensus how much this would be legal, but if seized, reparations may be available before the war is over. After Iraq’s invasion of Kuwait in 1990, $50 billion of Iraqi funds were seized and transferred through the United Nations to compensate victims in Iraq and other countries.
After receiving reparations, Ukraine will likely use them to rebuild the country. This may or may not include specifically paying back for individuals proportionally to the damage they suffered.
