1) Russia as we know it today was formed by Princes of Moscow. Moscow was originally subordinate to Vladimir-Suzdal which was the strongest in the region of Zalesye (today's Russia's core), but rose to prominence when there was a succession crisis in Vladimir-Suzdal ~1300. The Golden Horde, which collected tribute from the Russian princes, decided to issue a jarlig to them, a low-power actor, rather than the stronger Tver. Long story short, Moscow used this favouritism well and consolidated power, and as the Golden Horde's power faded by the end of 1400s used the position to not just annex the other Princes in Zalesye but also Novgorod the Great to the north. From there on the Russians fought against other Golden Horde successors (e.g. Kazan) and spread eastwards, at first in search and control of trade routes from Siberia, later as an imperial enterprise.
2) Kazakhstan (or rather, Kazakh Khanate/Ulus) formed after fragmentation of the Golden Horde. Originally the Kazakhs were part of the Uzbek Khanate under Abulkhair, but Janibek and Kerei, unhappy with his rule, collected a horde 200,000 men and migrated east, to the border with the Chagatai. Then there was a struggle for independence from the Uzbek in second half of 15th century which was, evidently, successful.