Basically, I understand you're referring to the conquest of the Siberian khanate by the Russian Cossack forces of Yermak Timofeevich. I just gotta point out that the khanate wasn't all of Siberia and the circumstances of the Russian expansion differed from place to place, but mainly there were two ways, one relatively peaceful and the other not so much. The conquest of the Siberian khanate, as it was the biggest and only actual state in the region, took a lot of years to be completed. In the 1560s the khans agreed to pay the Russian tsar the yasak (a natural tax) and that can be considered a protectorate, but really that was just a temporary measure because there was no real alternative for the khanate at the time, it was either incorporation in Russia or China later on. However in the early 1580s khan Kuchum rebelled against Russia so Yermak was sent to solve the problem. In a few years, mostly by 1588, the khanate had been annexed and the last Khan was taken to Russia for permanent residence, as Russian military outposts that later became cities were established in the area. So the garrisons maintained law and order, holding the territory together. That's the case with the Siberian khanate. In other cases Russia was dealing with small indigenous peoples that were mostly peacefully incorporated through the yasak and assimilation as Russia expanded and peasant settlers began to move to Siberia, creating the mostly Russian population that there is today. It's just a brief description because the process took more than a century with significant conflicts like the Dzhungar khanate invasions that destroyed a few Russian cities. To sum it up, there were two ways: 1) pay the tax and be peacefully incorporated 2) refuse and be annexed by force