I am not an expert on Kurds. Although I know more about their history than I did about Africa when I arrived in Ghana in 2011. So most of this post should be viewed as provisional thoughts by somebody from outside the field. Please feel free to leave any constructive criticism in the comments. The one element of Kurdish history I know a little bit about is the origins of their diasporas in Central Asia. Part of my PhD dissertation dealt with the nearly 9,000 Kurds deported by the NKVD from Georgia to Kazakhstan, Uzbekistan, and Kyrgyzstan in November 1944. Currently, I am slowly working on an article on Kurds and the USSR. It deals both with Soviet citizens of Kurdish natsional'nost' and Soviet assistance to Kurdish movements in Iran and Iraq during the Stalin era. The Kurdish diasporas in the USSR seems like a good place for me to start doing research related to the region.
Another topic I am interested in related to the Kurds is their role in the greater Afro-Asian project from 1958-1991. Obviously Kurds lived in a number of states active in the project including Kazakhstan, Uzbekistan, and Kyrgyzstan in the USSR, and Iraq, and Syria outside it. Turkey and to a lesser extent Iran under the Shah were US allies and hence outside Afro-Asia as a political unit. The fragmentation of Kurdistan between the rule of Ankara, Baghdad, Damascus, and Tehran presented an interesting problem. Kurds in Turkey were repressed by a regime clearly integrated into the US and European political bloc through its membership in the NATO military alliance. In contrast those in Syria and Iraq were repressed by regimes clearly part of the Afro-Asian project. These regimes espoused Arab Socialism, Pan-Arabism, and support for Palestinian national liberation movements. This led to a bifurcation of the treatment of Kurdish national movements by Afro-Asian states and the international Left.
Kurds in Turkey formed the PKK (Kurdish Workers' Party) a Marxist-Leninist guerrilla organization that got support from various Afro-Asian states and movements as well as European Leftists. In particular the PKK received support from the government of Syria and the PLO. They established a training camp in the Bekka Valley of Lebanon with the help of Damascus and the Palestinians. In this sense the PKK resembled other national liberation movements in Afro-Asia.
In Iraq the KDP (Kurdish Democratic Party) had initially been founded in Iran in 1946 with the support of the USSR. Until the early 1960s the KDP had good relations with the Iraqi Communist Party. But, that changed after 1961 as the KDP revolted against Baghdad and sought aid from the US and UK. Later after the Ba'ath coup in 1963 the KDP looked to Iran, the same regime that had crushed the Soviet backed Mahabad Republic where the party was formed. From this point on the Kurdish struggle in Iraq backed at times by Iran, the US, and Israel received very little attention from Leftist intellectuals and Afro-Asian states. In 1972 Baghdad signed a Treaty of Mutual Co-operation and Friendship with the Soviet Union. From this point on the USSR and its allies supported the Ba'ath regime against the Kurds despite the 1978 repression against the ICP (Iraqi Communist Party) and a complete break between the Communists and Ba'athists in 1979. In 1986 the KDP, PUK (Patriotic Union of Kurdistan), KSP (Kurdish Socialist Party), and ICP formed an alliance against Baghdad. The collapse of the USSR and the establishment of defacto Kurdish autonomy in Iraq both occurred in 1991. In contrast to the PKK fighting in Turkey neither the KDP or PUK received any significant support or notice from any Afro-Asian states or Leftist organizations in Europe or the Middle East.
The different treatment of the PKK in Turkey and KDP and PUK in Iraq by Afro-Asian states and European Leftists is notable. The cause of Kurdish national liberation in Iraq took a back seat to ideological considerations. The Arab center of Afro-Asia supported various strains of Pan-Arabism for most of the 1960s and 1970s. The Ba'ath strain in Iraq and Syria had no role for Kurdish national self determination. This exclusion of the Kurds of Iraq and Syria from support by Afro-Asian states and European Leftists is historically a major moral blind spot in these movements.