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 coincides to the proposed age of proto-NEC break up. (~5000 years ago) 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 also). Before the Kura-Araxes culture the Northeast 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. Another possibility is that the Lezgin-Udi group if languages formed on the southern slopes of Greater Caucasian range. In southern parts of Caucasian range Indo European tribes settled since the Middle Bronze Age period, while some others came in Iron Age. As Toponyms like Getaru, Gelaw can be related to those IE tribes.
Do all this mean that all Kur-Araxian culture was a NEC culture? Well not really because there is not much evidence of their presence in other parts of Kur-Araxian horizon. In other regions of KA horizon Anatolian and Hurro-Urartian terms are found. 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 Kura-Araxes was not a homogeneous culture as one might imagine looking on some maps on the Internet. Here I present a more precise map of Kur-Araxian culture where You can see that so called ETC (=Kura-Araxes) pottery on the Northeast of Caucasus is not classified as typical Kur-Araxian. That map is also remarkable because it demonstrates what we already learned from genetics that Kura-Araxes culture expansion had an uneven impact. In Levant and Malatya region it was very mixed while in South Caucasus it was homogeneous and dense.
This can mean that the pre-proto-NEC population do not directly descend from Kura-Araxes, it already existed in Late Chalcolithic and during the Kur-Araxian expansion the Proto-NEC community became part of that 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. Another possibility is that the Proto-NEC community was present in the proto-Kur-Araxians since the start. More ancient DNA will help to better understand this question.