Kotova addressed these possible cultural influences from the South:
Nadezhda Kotova, The Neolithization of Northern Black Sea area in the context of climate changes. In: Documenta Praehistorica XXXVI (2009). p. 164-166:
The Chokh culture itself was therefore not that influential for the formation of the steppe cultures.
Nadezhda Kotova, The Neolithization of Northern Black Sea area in the context of climate changes. In: Documenta Praehistorica XXXVI (2009). p. 164-166:
Some traits of Rakushechny Yar culture are similar to Neolithic sites in Eastern Anatolia: rectangular houses with daub, flat-bottomed pots, clay figurines,
polished tools, animal husbandry with domestic cattle, ovicaprids and pigs, but no horses. This similarity, together with close radiocarbon dates, allows
me to assume a borrowing of some attainments, or even a penetration of small groups of population from Eastern Anatolia to the Azov Sea area around 6900 calBC.
This migration could be the result of aridity, which has been fixed at c. 7000 calBC in the Azov Sea steppe (Bezus?ko et al. 2000.105). It was not a short
arid period, nor a local event. The transition from the Pre-Pottery to the Ceramic Neolithic has been recorded for this period in southeastern Anatolia. It was accompanied by a collapse of the Pre-Pottery
Neolithic cultures. Many sites were deserted.
[...]
The migration of some small groups of the Anatolian population along the eastern shore of the Black Sea was also possible. The similarity of the pottery found at the Chokh site in the Northern
Caucasus to pottery in Northern Mesopotamia, as recorded by Shnirelman (Shnirelman 1989.85), has confirmed this migration. Triticum dicoccon, Triticum monococcum, Hordeum vulgare and
Hordeum vulgare var. Coeleste; the bones of cattle and ovicaprids were found at this site, which is dated to c. 6900 calBC (Amirkhanov 1987).
[...]
Thus, for the second stage, two secondary centers of neolithisation are known in the south of Eastern Europe: eastern (in the Northern Caucasus) and western (in the Low Don region).
They mainly coincide with two variants of the Neolithic tradition as distinguished by Shnirelman, i.e., western, to which ? in my opinion ? the Rakushechny Yar culture is close, and eastern, represented by Chokh (Shnirelman 1989.85). The influence of traditions of the eastern variant has been not traced in the steppes of Eastern Europe, probably because their bearers occupied mountain areas. The steppe population of the Northern Azov Sea area appeared to have been more interactive. It is probable that the Early Neolithic of eastern Europe was formed solely under its influence.
The Chokh culture itself was therefore not that influential for the formation of the steppe cultures.