First I would like to point out that being part of the USSR is not the same as being a part of Russia. Russia itself was a part of the USSR.

Is your question specifically about the events leading to their independence from the USSR? Or is it about the "stans" national history and identity and the historical differences (and similarities) that exist with Russia?

It depends how far back you want to go. If you're only asking about their respective independence movements, then by and large no, there wasn't a similar process of popular protests for secession from the USSR. Many of the Central Asian countries, including Kyrgyzstan, Kazakhstan and Turkmenistan, sought to keep the USSR intact, even as voices to the west (including in Russia) sought greater autonomy or even outright independence leading up to 1991. For example, Kazakhstan only reluctantly declared independence from the USSR after its neighbours had already made this a de facto reality.

However, if you're asking about the origins of Central Asian nationalism, you have to go back further to both the Tsarist period and the early history of the USSR. I would be happy to expand on this if there is any interest.

When Russian forces conquered Central Asia, they conquered a multi-ethnic and multicultural region, where identity was based on familial, tribal and feudal lines. The concept of ethno-nationalism is a direct byproduct of Russian colonial rhetoric. After the conquest of Samarkhand in 1865 and the establishment of the Governor-Generalship of Turkestan in 1867, the Russian state used Turkic ethnographers to scientifically categorize their subject populations in the same way that geographers were mapping the region. To this end, these ethnographers looked for pre-existing divisions within Central Asian society, and to use these distinctions to establish definitive ethno-national groups of ethnicity in Turkestan.

One of the most important distinctions used by Russian ethnographers was that between the sedentary and the nomadic inhabitants of Turkestan. The settled, urban inhabitants of Central Asia were considered to be closer to Russians in the evolutionary lineage of civilization and progress than the nomadic groups. However, as Adeeb Khalid points out in The Politics of Muslim Cultural Reform, Russian thinkers were always afraid of the influence of Islam on Turkestan's inhabitants. They posited that, as Islam had become entrenched in value-systems of the urban inhabitants, it was the duty of the Russian colonial administration to sever the link between sedentary and nomadic peoples, in order to protect the 'noble savages' from the Islamic tendencies of the sedentary peoples. It's worth noting here that the term used for the nomads was interchangeable; 'Kyrgyz' and 'Kazhak' carried more or less the same connotation (you can see this in contemporary works on colonial ethnography, as in N. I. Ostroumov's Sarty). Olivier Roy's The New Central Asia describes the creation of two parallel systems of governance for the nomads and the sedentary inhabitants, and the destructive effects of Orthodox missionary work and an influx of Russian settlers on the societies of the nomads.

A further important subdivision within the urban populations of Turkestan was to found in language. The term 'Tajik' had long been used to refer to groups which spoke the Persian language (and kept many Persian cultural customs). By contrast, the Sarts (who would later become 'Uzbeks' under the Soviet administration) were an admixture of Uzbek conquering tribes, Turkic inhabitants and Tajik peoples. They were considered to be culturally and ethnically distinct from the Tajiks after years of their own cultural development. A more detailed analysis on the evolution and use of these terms can be found in Scott Levi's "Turks and Tajiks in Central Asian History" than I can hope to provide here.

Now, all of this being said, it would be wrong to assume that the Tsarist government ever hoped to exert extensive influence over the region. Imperial control was in reality very limited in Turkestan. Khalid describes Turkestan's colonial administration as constantly lacking in resources. Instead, the Imperial Russian state relied on a cadre of intermediaries, who established themselves as the new local elites of the colonial order, and often wielded significant power in their own right.

Such a setup was ideologically unpalatable in the early Soviet Union. Furthermore, a strong emphasis was placed on power delineation for national groups. As Francine Hircsh argues in Empire of Nations, Turkestan was broken up along with its administration, and the ethnic/cultural/national categories established under Imperial rule were used as a basis for the national territorial divisions we are familiar with in Central Asia today. These areas were afforded limited self-administration and governance. When the emphasis shifted from international revolution to socialism-in-one-country under Stalin, these territorial entities were physically entrenched, but on the basis of their continued participation in the larger Soviet project.

Edit: Combined my two responses for visibility.