Originally, the etymology of the Turkmen derives from Turk plus the Sogdian affix of similarity -men, and means “resembling a Türk“ or “co-Türk“, so all turks that were not in the classic Turk groups like Uyghur, Kyrgyz are all Turkmen. Turkmen now refer only to the Turkmen of Turkmenistan, Iran and Afghanistan, who speaks the Turkmen language. Turkmen are also affected by migration, whereby Turkmen took wife from Persia during the period of slave trade. Pure-blood Turkmen are groups who do not accept inter-marriage, like the Nokur in Kopet-Dag mountain range.
Turkmen are semi-nomadic people, so Mongol expansion was minimally invasive for the the Turkmen, with the exception of razing Urgench (now called Konye-Urgench meaning Old-Urgench), in what is considered to be one of the bloodiest massacres in human history. Despite the devastating effects of the invasion, the city was revived and it regained its previous status. It was Timur who put the final end to Urgench.
Turkmenistan today is in quite a different situation, governed by cult dictatorship and currently in its second iteration with Gurbanguly succeeded Niyazov. Turkmenistan can be termed “The North Korea of Central Asia”, where no independent visitors are allowed, so most travellers go with a 5-day transit visa, dashing into Azerbaijan or Iran from Uzbekistan. That said, Turkmenistan capital city Ashgabat is one of the most organized in Central Asia. Out of Ashgabat, conditions visibly deteriorate.