Baseyov is actually a bit more ambitious then that. He wants a Caliphate that stretches from the Caspian Sea, to the Volga Tartars and into Crimea and southern Ukraine. With him as master, of course.
The Abkhaz refugees in Tbilisi are actually Magrelian Georgians who were first settled in what is known as Magrellia (southern Abhazia and NE Georgia) by the Russian Tsar, along with Germans, as a buffer people to the Turks, who owned southern Georgia at that point. Some of these people have returned, but they often get preyed upon by their own people who are called "partisans" (supposable fighting the Abhaz) but who's more accurate name should be Mafia.
As for Tbilisi, the refugees are actually now the majority of the population. Those crowds of Tbilisians who were marching in 1991 screaming for independence from the Soviet Union and Moscow in particular are all gone. Where to you might ask? Easy, to Moscow. Yup, the big irony is, they thought their fantasy economy, made possible solely by the idiotic Soviet economic conditions (the very thing that brought the SU down) would continue once they were independent. They thought their factories would keep running, never mind that they have ZERO natural resources, no gas, oil, or coal or the fact that those factories produced crap that no one wanted (except the Politburo's central economic planning committees) and that the products sat in warehouses for decades. Once they figured out the mess they made for themselves, these protesters were the first to pull up roots and head to Moscow or Europe, about 1 million now live in Russia. The present people of Tbilisi are mostly village people from the mountains. Why that city is such a wreck.
You think? I fell madly in love with it, minus the hungry strays on the streets.