To start out answering this, I would caution against leaning too heavily on generalizations, because we are ultimately dealing with an extremely small data set, and there are some big exceptions to the generalities. Belarus has very notably been a fairly repressive dictatorship since Alexander Lukashenko's election in 1994 (I think we can safely assume he will be re-elected for his sixth term next month). Conversely, for all of the genuine instability it has suffered, Kyrgyzstan has had a relatively pluralistic political system with a fairly open press.

Still, there is some truth to the generality, and it's not even something related to the post-Soviet era: Boris Yeltsin preferred jettisoning the entire Union rather than be outweighed in a possible surviving system where the conservative, authoritarian Central Asian SSRs were not balanced out by more liberal European republics (with Ukraine being the clincher). So what gives? Oriental despotism? The allure and dream of European values? The inherent violence of Muslim societies? Lord only knows writers of various influence have made these arguments.

Anyway, I'm going to throw those arguments out. Instead I will argue that what we are seeing is the end result of decades of Soviet nationalities and economic policies, plus a healthy dose of historic contingency.

Regarding nationalities policy: for Central Asia, the current political entities there are relatively new, and date from the 1924 territorial delimitation carried out under Stalin's auspices as Commissar of Nationalities. Now, this doesn't mean that national identities there are new, or even more that they are "artificial" - despite lots of local haggling, the borders as drawn did accurately reflect general ethnic boundaries. What was happening in Central Asia is that national ethnic based identities were not paramount in peoples' self identification: someone who is Turkmen would have a sense of being such, but probably be much more likely concerned about which tribe and clan they were a member of. People who were farmers and without strong tribal identities would often be considered "Sarts", in contrast to various nomadic peoples. Religions cut communities into sub-communities, multilingual people would often self identify with either or both of their languages based on trends, and people would just as likely identify themselves with their mahalla neighborhood community (in Uzbekistan) as anything else. The exceptions in this matrix would be a nascent Kazakh nationalism in the Slash Orda movement, which was influenced by the struggle against settler colonialism on the steppe, and the jadid movement, which was an Islamic modernization movement influenced by trends elsewhere in the Islamic world in the early 20th century, and found most of its prominent members in the 1920s among the Young Khivan and Young Bukharan parties (Khiva and Bukhara were former tsarist protectorate khanates that became allied People's Republics of the Bolsheviks, before being diseastablished and absorbed by the USSR in 1924.

When Soviet control was established over the region in the 1920s, the Bolsheviks faced major governance problems. The Bolshevik party itself in Central Asia had been mostly comprised of European settlers to the region, who lived separate from "natives" - the two groups were incredibly hostile to each other. Local Central Asians often were connected to widespread basmach "bandit" movements (the Bolshevik term - we would more properly say insurgent groups). These groups occasionally had notable foreign support, both from compatriots over the border, and even from figures such as the former Ottoman leader Enver Pasha, who sought to lead a pan-Turkic movement and was killed in modern-day Tajikistan in 1922.

The Bolsheviks ultimately were able to prevail through pure military power, but faced a conundrum over how to get local buy-in to the new regime. The first major effort was to coopt local reformist forces, like Alash and the Young Bukharans. Many members of the latter even joined the Bolshevik party, but there was always a fear that these party members weren't really on board 100% with whatever the party platform was.

Thus we get to the policy of korenizatsiya, or "rootification", namely to build local communist cadres that were brought up and loyal to the system, but that also had a relatively widespread sense of legitimacy. This meant, as seen, that territories were delineated around broadly ethnic lines in 1924, and the units so established were supposed to promote the "titular" nationality of their area: the Uzbek SSR was to use a (standardized) Uzbek language and culture, and favor the development and promotion of Uzbek party elites. Moscow policy would waver in its commitment to korenizatsiya, and in the 1930s purges saw most local national leaders arrested and shot as "bourgeois nationalists", with Russification in language, culture and local personnel becoming more favored, but the principal remained the same: a republic was supposed to promote a particular national identity, defined through an officially-sanctioned language and culture, and promote development that was "national in form, socialist in content".