Good question, but surely one would want to bring Indo-Iranian into it. (Which nobody ever does for anything involving IE, with the exceptions of Sanskrit and Avestan.)
From
Encyclopedia Iranica, it sounds like the Iranian languages replaced the dative with the genitive and extended the use of this new genitive-dative to a general oblique function. Earlier stages of Iranian with more case contrasts are attested - in particular, Yaghnobi has the common Iranian direct-oblique system, but its apparent ancestor, Sogdian, still showed signs of a six-case system.
The replacement of the dative with the genitive is also seen in Tocharian. (
And it's considered to be a feature of the Balkan sprachbund.) Which Indo-European cases do the Tocharian languages continue? The sum of TA and TB cases is:
- nominative, continued from PIE
- vocative (TB only), continued from PIE and lost in TA for entirely phonological reasons
- genitive, continued from PIE and partially replacing the dative (possibly there was an earlier stage where the genitive
thoroughly replaced the dative before the TB expansion of the perlative), although the origin of TA GEN.PL -śśi is unclear. Considering the full paradigm in both languages - TB -ntse, -nts(i) < *-nesos, *-nesom, TA -s, -śśi - it's possible that this is another piece of evidence for *dʲ > TA ś, TB ts.
- accusative, continued from PIE
- instrumental (TA only), of unclear origin (the form -yo suggests *-yaw / *-yow)
- perlative, postpositional in origin and reconstructible to PToch, from *ā with -s- in TB due to metanalysis in plurals: *-ns-ā > *-n-sā
- comitative, of unclear origin but with non-cognate TA and TB forms, so not reconstructible
- allative, postpositional in origin and reconstructible to PToch, from *de according to Adams but wouldn't this be phonologically irregular? Early TB -śc, TA -c - the TB sibilant can be explained with metanalysis parallel to the perlative (and later TB -ś is a phonological development), but the presence of -c in both languages suggests instead *-(t|dʰ)(e|i) - *de should give -ś, or TA -ś, TB -ts, or something like that.
- ablative, inherited from PIE but non-cognate! According to Adams, TB -meṃ is "from the PIE ablative plural *-mos, rebuilt after the accusative plural as *-mons", and TA -äṣ ( / -aṣ / -āṣ) is from *-d/ti and cognate to Hittite ablative -z, but this looks phonologically irregular - ṣ should be from palatalization of *s, not of *t or *d
- locative, postpositional in origin and reconstructible to PToch, from *h1en
- causal (TB only), actually a rare and entirely fossilized adverbial formulation rather than a case, but possibly continuing the PIE instrumental
So PToch likely continued the PIE nominative, vocative, genitive, accusative, and ablative, and possibly also the instrumental, although it could've been fossilized even then. (This is in principle something it may be possible to gather evidence for or against - the "causal case" only appears on six words, so their etymologies could be examined. The causal would've had to have been productive at the time the words came into common use.)
I don't know what happens in Indic.
But the case systems of most of the "classical" IE languages were already somewhat reduced - excluding the vocative, Greek had four cases and Latin had five, whereas PIE is reconstructed with around seven. So it seems reasonable to say that most of 'Brugmannian' Indo-European had a tendency toward loss of cases, which was arrested in Tocharian and to some extent northern Balto-Slavic due to Uralic influence. The real question is why
Proto-Celtic (allegedly) continued so many cases for so long... maybe that's where the Basque monks come into it.