IE *h₁ek´w-o- (Latin equus, Greek híppos) 'horse'1 is surely a Wanderwort whose earliest form can be traced to Nakh-Daghestanian *ɦɨ[n]tʃwi (~ -e) 'horse' (NCED 211) and which originated in the Pontic-Caspian steppes, where the animal was domesticated around 4,000-3,500 BC2, and also spread to Sumerian anše 'donkey' and Hurrian eššǝ 'horse'3.
Possible correspondences are Afrasian *ʕi(n)ʒ- 'sheep, goat' (Semitic, Cushitic) and *χu(n)ʒ(-ir-) 'pig' (Semitic, Chadic)', whose specialization could have involved phonosymbolism in the initial fricative.
Taking into account Uralic *ki(n)tʃe/*ky(n)tʃe 'nail, fingernail, claw', my bet is this particular word would have first been originally to Pleistocene ungulates. As pigs, sheep and goats -which together with cattle made up the Near East Neolithic package- were first domesticated around 9,000-8,000 BC in the Taurus-Zagros mountains area, I would identify the source language with Bomhard’s Proto-Nostratic4, which his collegue Kerns5 regarded as being part of the Dene-Caucasian phylum: I believe that Nostratic languages did not exist except as a part of Dene-Caucasian until the waning of the Würm glaciation, some 15,000 years ago.
However, in my opinion this language couldn't be the common ancestor (Mother Tongue) of a plethora of language families such as Indo-European, Afrasian or Kartvelian, but rather the source of several Neolithic Wanderwörter.
1 In fact, the IE word has been used by defenders of the so-called Kurgan theory as part of the evidence supporting those people were speakers of PIE (i.e. the proto-language of the IE family). See J.P. Mallory (1989): In Search of the Indo-Europeans. Language, Archaeology and Myth, p. 143-185.
2 The domesticated horse (Equus ferus caballus) is a different subspecies than the wild horse of the Eurasian steppes (Equus ferus ferus), also called tarpan (a Turkic word). There is also another horse subspecies native to the Eurasian steppes, the so-called Przewalski's horse (Equus ferus Przewalski's), which has never been domesticated.
3 Luwian *aššu-/*azzu- 'horse' and Georgian aču/ačua 'interjection for calling horses' are loanwords from Indo-Iranian.
4 A. Bomhard (2008): Reconstructing Proto-Nostratic. Comparative Phonology, Morphology, and Vocabulary, vol. I, p. 235-241.
5 A. Bomhard & J.C. Kerns (1994): The Nostratic Macrofamily: A Study in Distant Linguistic Relationship, p. 153.