Nakh languages have some similarities with IE. Concerning Chechen, for example.
The ergative of the 1.SG pronoun is /ʔəs/ and we have nominative 'es' / 'as' in many IE languages. (like Lithuanian, Armenian)
The 1.PL pronoun is /vəɪ/, fairly close to how it is reconstructed for IE by Beekes and Sihler *uei / *wei
The genitives of the pronouns have an -n (sən, ʜən, vəɪn, txʰən, ʃun). An -n exists in the reconstruction of the genitive of the IE first person pronoun (*mene, h₁méne) and Uralic first and second pronouns (*mun, *mina 'I'/me, *tun, *tina 'you')
The 3.SG singular personal pronoun is ɪ, ɪzə, which can be related to IE demonstratives or pronouns. For example a Latin/proto-Italic equivalent is *is from an IE pronoun reconstructed as*éy.
The plural is often formed by adding suffixes -ii or -ash which are at least reminiscent of IE suffixes.
k'ant (boy) - k'entii (boys)
ph'āgal (rabbit) - ph'āgalash (rabbits)
Example:
Latin lupus - lupī but for example dux - duces
Attic lukos - lukoi but fore example rhetor - rhetores
Sanskrit vṛka - vṛkas, OCS vlьcŭ- vlьci, Lithuanian vil̃kas - vilkaĩ etc
Although both Nakh and Dagestanian are considered NEC languages, the relationship is considered distant despite the geographical proximity. Either way, I don't claim there is a close relationship with IE or even that I can prove that any relationship exist (or even that I want or plan to do it, I don't have the necessary free time and motivation)
Either way, I believe that might be explained with the following scenario
-pre-proto-Indoeuropean possibly in Catal-Hoyuk (something Diakonoff had considered but it is an assumption with 'no necessary validity' as he had said)
-a movement of a pre-IE / para-IE group to North Caucacus (Maykop), responsible at least for the similarities between IE and North West Caucasian (Adyghe, Abkhaz)
-late Indoeuropean possibly in the Balkans, Hungary, West Ukraine.
-late PIE expansions towards east
1) first wave with Globula Amphora like ancestry
2) second wave with Steppe MLBA like ancestry (Sanskrit speakers could have followed a southern route, though. I don't consider the conclusion of the S. Asian study necessarily valid)
-proto-Uralic languages developed in proximity to two IE languages, a Baltic one spoken in Fatyanovo and an early Indo-Iranian that had influenced Abashevo culture and it expanded mostly the last 4.000 years.