The short answer is yes: the Russian Empire did have settlers from its European parts move in large numbers to new regions, especially in modern-day southern Russia/Caucasus region, and also in Central Asia and Siberia. By the last decades of the 19th century and first decades of the 20th, this was official government policy and in explicit emulation of the American West. The idea was to relieve overpopulation and "land hunger" among the rural population in the European portion of the empire by offering land for new farms. About five million settlers moved to Siberia and a million to Kazakhstan to farm new(ly appropriated) lands in the two decades prior to World War I, and the Soviet development of Siberia (infamously but not exclusively through Gulags and "Special Settlements") was a continuation of this process.

u/kaisermatias talks about the Caucasus and I talk about Central Asia here, and I have a follow-up on Siberia here.