When I began studying the interaction of Uralic and Turkic languages in the Volga-Kama area, I assumed that existence of a feature in both Tatar and Kazakh was sufficient to prove that Tatar inherited it from Common Turkic and did not borrow it one of the languages of the Volga-Kama area. However, it turns out that contact between the Kipchak languages persisted long enough for North Kipchak to contribute some loanwords to South Kipchak.
The first example is Kazakh moncha ‘sauna’. According to Klára Agyagási in Ранние русские заимствования тюркских языков волго-камского ареала Ⅰ (Debrecen, 2005) p. 58, this ultimately derives from Russian баня, borrowed into Ancient Chuvash with the rounding of a typical for early Chuvash and the shift of b > m before nasals typical for Turkic in general, and finally taken up by the South Kipchaks sometime before the North Kipchak vowel shift (cf. Tatar munča).
The second example comes from a paper by András Róna-Tas, “Three Volga Kipchak Etymologies” in Studies in Chuvash Etymology I. (Szeged: Szeged University Press, 1982). He traces Tatar and Bashkir izge ‘holy, good’ back to a Volga Bulgarian form that produced modern Chuvash ïră. Kazakh izgi ‘kindly’ must therefore be a loanword from the Volga Kipchak languages.