One of the complexities of Armenology is that it is not sufficient to learn just Armenian related stuff but You also need to understand all neighbouring populations history. So let's see North East Caucasian (Nakh- Daghestani) people origin and dispersal.
If we exclude that young cluster of J2 among Nakh people the common denominator of all NEC people is the J1-Z1842 branch. More specifically the two subbranches of CTS1460. The coalescence age of this branches coincide to the proposed age of proto NEC break up. (~5000 years) so we can assume that those are the genuine NEC lineages. The origin and the expansion of Z1842 is usually linked with Kur-Araxian culture ( until now ancient DNA supported that idea, but Late Chalcolithic expansion can't be ruled out ) . Before the KA culture the North East of Caucasus was sparsely populated place and only with the advent of that culture sufficient population appeared in that region to disperse.
The biggest linguistic diversity of NEC languages is found in the North while in the south we have only the Lezgin-Udi group, which might have migrated from the North at LBA/IA. They settled in the Northern parts of Kur river and later were known as Aluank' / Albania kingdom. But in most likelihood some IE elements were present in that region from older periods. As You can see from map some terms like Getaru, Gish, Gelaw can have IE origin.
Do all this mean that Kur-Araxian culture was a NEC culture? Well not really because there is no much evidence of their presence in other parts of Kur-Araxian horizon. Then how to explain their connection with Kur-Araxian lineage and culture?
This is somewhat complicated but in most likelihood it has to do with the fact that KA was not an homogeneous culture as one must imagine looking on some maps in the Internet. Here I present a more precise map of Kur-Araxian culture where You can see that so called ETC (=Kur-Arax) pottery on the North East of Caucasus is not classified as typical Kur-Araxian. That map is also remarkable because it demonstrates what we already learned from genetics that KA culture expansion had an uneven impact. In Levant and Malatya region it was very mixed while in South Caucasus it was homogeneous and dense.
In sum we can deduce that proto-NEC population do not directly descend from Kur-Arax but has borrowed from them the material culture and then dispersed on its own with subsequent cultures like Guinchinskaya (MBA) and Kayakent-Kharachoy (LBA/IA) which is also found in north Azerbaijan.
The next question that arise is then who were those mysterious Kur-Arax people and their J1-Z1842 lineage. Keep in mind that Z1842 is just one of their lineage they were probably other also.
Well many scholars believe that KA folk were Hurro-Urartians ( we can tentatively add Kassites also ) some others proposed Anatolians but maybe You would have a more interesting theory. :)