A question post in the comments: how plausible is that much of South Asia today is dominated by languages that were introduced agro-pastoralists in the period between 2000 and 1500 BCE? (the proposition put forward in The formation of human populations in South and Central Asia). This is not a peculiar question. It seems that in the range of 10-20% of the ancestry in the Gangetic plain derives from people whose origins are in the Eurasian steppe (please see the table at the bottom of the post, which I derive from Narasimhan et al., for why I say 10-20%).
Is it plausible that agro-pastoralists from the Eurasian steppe gave their language, their mythos, to the Indian subcontinent, when most of the ancestry predates their arrival? If the figure is 10-20%, I would say it certainly is plausible and entirely within the realm of likely given the model (male-mediated expansion).
Some non-Indian examples.
The Ugric languages are widely dispersed in Eurasia. The reason being that the Hungarian language is a recent cultural artifact on the Pannonian plain. Genetically modern Hungarians have only trivial (1-5% most generously) affinities to Central Eurasia, being genetically similar to their German and Slavic neighbors. Ancient DNA has yielded that Hungarian nobles from the medieval period were more Asiatic in the background, with exotic Y chromosomes. One explanation for the discrepancy with moderns is that the flower of Hungarian nobility was decimated by the Mongol invasions, as well as the latter predations of the Turks. But, in any case the Magyar and their Turkic federates were never likely more than 10% of the population of the Pannonian plain.
(in contrast to the Magyars the Avars and Bulgars left no linguistic imprint, and were assimilated into their substrate; the Bulgars only leaving their ethnonym despite being Slavicized)
Before the arrival of the Angles, Saxons, and Jutes, the people of the British Isles either spoke a Brythonic dialect or Latin. Were the Anglo-Saxons a major demographic shift? Or did the switch to Old English, which is really a German dialect, happen through elite transmission? The best work today suggests that at most 25% of the ancestry of England proper is German, with perhaps a high fraction of 40% in East Anglia. The switch to a German identity in the centuries after the 5th-century withdrawal of Roman legions seems to involved both migration and assimilation into the barbarized identity.
In the late Roman period, the dominant language in the Balkans was Latin, with Greek and Illyrian minorities. In the 7th-century the Balkan limes collapse. Today Slavic languages are dominant (Romanian and Albanian being exceptions). Though a significant genetic impact occurred through the Slavic migrations, it looks to be less than 50% in the Balkans. The barbarian Slavic tribes assimilated Latin peasants into their culture.
The Turkic languages have expanded a great deal over the last 2,000 years. Today large swaths of Central Asia that were once Iranian speaking are now Turkic, while Anatolia, which was once Greek, Iranian, and Armenian, is now mostly Turkish in speech. In the eastern edge of Turkdom East Asian ancestry is predominant. But once one reaches Anatolia the fraction is the 5-20% range.
Arabic has famously spread outside of its Arabian-Syrian homeland since the rise of Islam. I think it is difficult to judge the genetic impact in the Fertile Crescent because these Aramaic and Assyrian speaking populations had long been interacting with north Arabian peoples. That being said, a comparison of religious minorities (e.g., Assyrian Christians) with their Muslim neighbors indicates that only a minority of the ancestry is due to migration since the rise of Islam, though in some areas it may be substantial (e.g., 30-40%).
A better test is Egypt, where the linguistic substrate (Coptic) was significantly different from the Semitic of the Fertile Crescent. It looks likely that there is overall continuity with Pharaonic Egypt, though some Levantine, Turkic, and Sub-Saharan African ancestry has been added. In contrast, in the Maghreb Arabicization has been overwhelmingly cultural.
Latin. Iberia and Gaul adopted Latin through cultural processes. The genetic impact seems marginal (the same is true is the Balkans, but there only Romanian remains of the Latin period; though Romanians show no greater affinity to Italians than their neighbors).
The Chadic languages are a curious branch of Afroasiatic languages in West Africa. Less than 5% of the ancestry of these people is West Eurasian.
Spanish and Portuguese in the New World. Substantial genetic impact. On the order of 50%. But far less in some areas than others.
The Australian Pama-Nyungan languages seem to have spread mostly via cultural diffusion or some sort, rather than genes.
About 30-40% of the ancestry in Iberia is steppe. But the people seem to have spoken Indo-European langauges (though some not!) before the Romans. In Italy 30-40% is also steppe, dating to the same period. Again, Indo-European languages. The fraction in Greece is lower for the antique period. Again, Indo-European languages.
In China, Japan, and Northern Europe, language shift seems to have been accompanied by preponderant demographic shifts (though not overwhelmingly or exclusive). The shift from hunter-gatherers to agriculture saw the same in Southeast Asia, though the shift between language families is often mostly cultural (e.g., Thailand is only minority Tai in ancestry).
The moral of the story is that it is possible that unlettered barbarians can impose their language and culture on a people more advanced, but fallen. Ask the Britons. Ask the Greeks of Anatolia, or the scions of the Pharaohs. Ask the Minoans and the Pelasgians. The men of Urartu who fell to the Hayk.