I dont know where to start, I found that my family tree ends with a "freed slave" in Finland, I assume the owner was a Russian because the surname is Russian. The year is 1780 with the relative having an age of 35. (If that would help paint a picture.) Käkisalmi is the Township where the Church record is found , listing my ancestor as a freed slave. Surely the Fins did not own Russian slaves? Be nice to me... Its not everyday you find out your ancestor was Spartacus and not Caesar.

The simple technical answer is: there was no such thing. Slavery did not exist in Sweden nor did any other bonded labour such as serfs, and consequently it didn't in Finland either. There were no laws covering it or any real information because basically it didn't exist. Of course history isn't easy, there existed a Swedish slavery (and slavetrade), but only in a limited context. In 1784 Sweden was granted a colony in the West Indies, St Barthélemy, by the French. It came with slaves, slave trade and the problem that there wasn't really rules to cover any of it's activities under existing Swedish law. Basically the colony was ruled by a different set of rules (much of it copied form surrounding Carribean island's practice and laws) than Sweden proper (and thus later Finland). In short in the context you provide slavery does effectively not exist.

No Swede (and therefore Finn, same thing at the time) would own a slave, in Sweden, and absolutely not someone who would be seen as white, like a Russian.

Now, Käkisalmi and the larger administrative unit it belonged to would in 1780 actually not be part of Sweden (Finland). So yay slavery is back on the table...

In 1721 in the peace of Uusikaupunki/Nystad the parts that would later be the Käkisalmi lääni was ceded as a large part of the Viipuri/Savolax lääni to Russia along with the Baltic provinces. Another part of the eastern Swedish areas were ceded after the failed war in the peace of 1743 and these were all organised into "Old Finland". In wasn't until 1812 after the rest of eastern Sweden now termed the Grand Duchy of Finland (an old name that hadn't been in use for a century) was all gathered into a single administrative unit by the imperial Russian administration, at which point all "Finnish" areas were part of the same "nation", Finland technically wasn't Russia, it was a personal domain of the Grand Duke who happened to be the tsar of Russia. As part of the transfer of rule in 1809-10 it was stipulated that the Swedish law in force would remain in force in Finland. Which it more or less did until 1919 (although it was modified a fair bit, at it's heart it still was the Swedish Law of 1734, plus some later absolutist amendments). This also led to some issues in the parts of former "Old Finland" as it was unclear whether Russian laws or the Swedish ones would apply in the areas. This had implications amongst others in the application of serfdom which of course was the legal principle of most peasants in the Russian empire whereas Swedish law did not know the concept. Finnish landowners naturally (along with Russian ones, many Russian nobles were granted "fiefs" in the area as it was conveniently close to the capital of St Petersburg) weren't always keen on reading the agreements as generously. Originally they were only to collect the taxes the peasants were due to the state, but slowly the peasant's rights were eroded towards serfdom. This served as a powerful motivator for the rest of the nation to not easily accept increased Russian suzerainty. And broadly speaking all through the period of Russian rule the Finnish were desperately holding back attempts of expanding imperial control and "Russification" by trying to establish principles of legalism and adherence to the original agreement the tsar agreed to. Relatively successfully too all things considered.

Slavery proper was abolished by tsar Peter the Great in Russia in 1723 too, most peasant-slaves having been "converted" earlier. Only to be replaced by serfdom, which in reality seems to have been more of a paper product. That is to say things carried on mostly similarly under a new label. In effect it does seem Russian "owners" basically could treat "serfs" more or less equivalent to the chattel slaves of the Americas, rules or no. Which sort of answers your question how they were made to work. With threat of physical violence and sale. Mind, serfs working land usually had titles to use the land for life so kinda had incentive to stay there, and without no real possibility to better oneself being a landed serf had some certainties of life. I'm no great expert on Russian slavery/serfdom though. There could definitely be someone better informed here.

So to try to make some sense of this, which unfortunately is a mess of overlapping things that connect in complex ways.

Your ancestor was most likely a former serf not a slave. In practice the difference could basically be negligible though. Now people did get sold akin to slavery, Russian nobles even illegally exported their serfs to be sold into Ottoman slavery. Also descriptions of the treatment of prisoners of war hits close to slavery as well. And again the option existed of simply selling people into slavery. Finnish history talks about people being taken during the wars of the 1700s "as slaves" by the Russians. Few returned. But I can't speak of what that would in practice end up as. I assume many were sold elsewhere or effectively put into serfdom somewhere in the vastness of Russia.

Finns probably didn't have many Russian serfs no. Most likely some Russian noble would have received the "fief" (I don't know how all the bits were distributed at that time) your ancestor could have belonged to between 1721-1812. Most such lands would have had self-owning farmers when it belonged to Sweden and it wasn't until the 1810 hand-over that top-level owners would also been transferred, that is to say no top-level landowners would be included in the transfer so the people would be self-owners or crown tenants. The inhabitants shouldn't have been either slaves or serfs, but the conditions tended to slip towards that. As a "former slave" it's quite possible your ancestor was actually transferred from elsewhere and planted upon one such new "fief" a Russian noble would have been granted by the tsar for their conduct in the wars. It is also not necessarily directly possible from the records based on a name to ascertain ethnic/national/linguistic belonging. Like say you start out as a Finnish speaking peasant, get conscripted to the Swedish army and get enrolled with a Swedish name, become a prisoner of war and end up as a serf in Russia only to later be moved with your landlord to Käkisalmi and there be registered under a now Russian name.

Since slavery nor serfdom technically didn't exist in Finland there aren't really any records proper covering such that I can think of. There could be various types of legal documents if your ancestor might have reason to process against someone. But there is also the problem that in Finland proper there was a rule of law, whereas the areas which effectively were ruled as fiefs I don't see much such paperwork existing. The landlord would decide what was the law and that might not generate much paperwork.

Also, another way of looking at it, is that I'm not quite sure how exactly we construct what a "Russian slave from Finland" actually is, in the context provided in the question, and I have tried to extrapolate from your additional post.