I'm definitely not a linguist, but certain correlations between language families and results make for interesting speculation. Language and genetic affinity obviously correlate only imperfectly, otherwise the Iberians wouldn't speak a language from Latium, Irish wouldn't speak mostly English nowdays, Turks a language from the eastern Central Asia region, and Egyptians not Arabic. Still languages generally correspond to at least elite substitutions. However during the Neolithic vast demographic changes apparently did take place, and these were necessarily accompanied by linguistic replacements.
The correlation between the genetic expansion of the West African Neolithic and the Niger-Congo languages seems quite straightforward. There are sufficient indications in my opinion for certain older or newer theories of language affiliation to perhaps be revisited? I'm a blogger so I'm allowed to make outlandish linguistic claims using genetic data without presenting new linguistic evidence. All these theories have been presented before, and generally rejected for lack of evidence. But it's difficult to find common roots going back 10.000 years (unlike Indo-European which is more recent). Here it goes. I'm obviously not particularly confident about any of the following theories, still a Neolithic origin for current language families does make sense.
1. Elamo-Dravidian: Elamo-Dravidian, a proposed family in the past, perhaps also with Sumerian? seems perfectly correlated with an early expansion of the Mesopotamian Core peoples to the East towards India. It was perhaps associated with Southern branches of the R1a Y-haplogroups. Maybe a distant connection with Indo-European through the Alarodian family?
2. Vasconic: this highly controversial language family, rejected by the current consensus in the area, actually has some genetic backing, since Proto-Vasconic's expansion could be tied up to the early Mesopotamian Core expansion towards the Mediterranean and Atlantic and perhaps also (or a parallel branch) into the Balkans and Central Europe. If true, the ancient Megalithic people would have spoken these languages up to Scandinavia. Associated with Y-Haplogroups R1b.
3. Afro-Asiatic: in my view the home of proto-Afro-Asiatic must be the Nile Valley, or at least its posterior expansion was from there. It would correspond to the Nile Core second wave.
4. Indo-European: Proto-IE may have been spoken in the Kurgan area. It expanded with pastoralism into Europe and India. The Proto-Germanic-Italo-Celtic branch would be born over a heavy Vasconic substrate in the Corded Ware melting pot, expanding from there with mounted warrior bands and perhaps also the contemporary development of Rye agriculture, allowing for a second "Eastern" Mesopotamian Core expansion due to the difficulties of Wheat/Barley agriculture in many Northern European plains soils versus Rye. Indo-Aryan would be the natural evolution of original IE spoken in the steppes, and it's affinities with Balto-Slavic explained as these being the most conservative, less substrate-affected branches.
5. Caucasian languages: mountain peoples are generally spared the largest impact of demographic waves. These could correspond to surviving very old Fertile Crescent-Mesopotamian Core languages, maybe with some relation to the Vasconic, Indo-European and Elamo-Dravidian families.
6. Niger-Congo is straightforward as said.
7. Nilo-Saharan tongues may be derived from the expansion of Eastern Green Sahara pastoralists around 3000 BC?
8. Japonic languages: I will present East Asian data shortly, but it seems to me that these languages were the ones spoken in the original Yellow River Core peoples expansion. Some relationship to Altaic as these were carried to the steppe by the development of pastoralist lifestyles at the periphery? Maybe Chinese proper later arrived from the Yangtze River Core with Rice.
10. Iberian languages and Tyrsenian languages (including Pelasgian?): Vasconic-related tongues? Afro-Asiatic derived? Hybrids?
Tongues change faster than genes. Elites can and do often impose tongues on peasants alien to them. However when discussing the origin of families and their ancient spread, the Neolithic Revolution's genetic impact may offer some clues.