“Turkey MAY want to get the Caucasian lands, but can they?”
More likely they would exercise soft power to support independence of Turkic peoples from Russia (and closer to Turkey). Political and diplomatic support in International relations; covert support and agitation.
Although the Turkish economy is a basket case, they have one of the top ten defense industrial bases on Earth (Turkish companies got the contract to expand US artillery shell production in Texas). Turkey also exercises a lot of soft power and influence in Muslim Central Asia, with popular media (TV, movies, music), as well as religious and cultural outreach programs.
Very good points. I completely forgot about the spread of Turkish television, which is very popular in its near abroad.
You are correct, they may be able to just walk in without firing any bullets