Are you referring to the R1a/R1b problem? We do know that non-Corded Ware-derived Indo-European speakers like the Greeks, Armenians, Albanians (the latter perhaps derived from ancient Illyrians) all carry significant amounts of R1b, like the Yamnaya. Why Corded Ware carried R1a instead is still an open problem but not fatal to the hypothesis that Corded Ware originated in a Yamnaya migration (e.g. there might have been different elite paternal lineages involved that weren't dominant enough to be represented in steppe kurgan burials).

Haplogroups aside, the replacement over thousands of years is the most interesting detail to me.

In Iberia, you have northern Iberian groups which maybe spoke a bit of an Indo-European language at first, and they are invading Southern Iberians and leaving almost no trace of other haplogroups.

Britain is less interesting because basically it was a regular colonization/invasion of the kind we know by Britons; although there is a genetic trace left of the megalithic people, they were swamped.

As a contrast to the Iberian case, we have several steppe invasions in the historical record but most of them look nothing like it.

The Khans did expand rapidly and engage in some wild oat seeding but there is barely any evidence that they ever were in most of the places they conquered. The Huns were basically new management for Germanic tribes. Again, no trace of their existence.

The only one that looks a bit like the Indo-European expansion is the Turkish invasion of Anatolia where by the end you have a bunch of South-Eastern Europeans with minor Asiatic ancestry effectively larping as Islamic Asians and even becoming the center of the Islamic world for centuries.

But, even in the Turks, you don't see full Y-DNA replacement, not even with the power of Islam.