David Anthony, author of The Horse, the Wheel, and Language, now the only book you really need to come up-to-date on Indo-Europeans, has new research, but his next book, The Dogs of War, won't be out for three years. In addition to explaining new understandings based on the human DNA explosion of the last decade there will be info on horse DNA related to domestication, and a description of what is iup with the dogs. The dogs (and sometimes wolves) at Yamnaya initiation sites, that is, that seem to have been eaten in ceremonies in preparation for cattle raiding. It is notable because they did not consume dogs at any other time of year. The likely interpretation is that this was somehow about becoming dogs or wolves, which concords with early mythologies with Indo-European peoples.
To review. The Yamnaya domesticated the horse some time before 3000 BC. They may have gotten the idea from the Botai, a nearby tribe, but those horses are unrelated to current horses, while the Yamnaya horses are the direct ancestors. So also with the wheeled wagon, which was possibly also developed in the North Caucasus about this time, but seems independently invented by the Indo-Europeans. So they may have seen them or heard about them beforehand, but built these heavy clunky suckers all on their own. This allowed a mobility heretofore by any other tribe - not anywhere.
Prior to this there were deeply related people living in the river valleys such as the Dniepr and Volga. They hunted, fished, and had some agriculture in the narrow bands of forest on either sides of those rivers. Most especially they had cattle, sheep and goats, and hunted wild horses. But once the horses could be ridden and all the necessary goods of survival could be loaded onto wagons, the vast grasslands between the rivers could now be exploited. They had been previously empty of humans, but now there was unlimited food for animals and the humans moved in. They covered enormous distances rapidly. There is a Yamnayan individual found in the Altai mountains and another very close in era in East Slovakia, 4000 km apart. They are fifth cousins, sharing a common ancestor 150-200 years before. They were the first instance of those steppe cultures that persisted for thousands of years.
The river settlements emptied except for a very few in what is now Ukraine. These pastoral nomads built no new settlements of their own, because they did not need them. They lived permanently on the move. What they left behind was their language and their genes. They did not merely wander. They seem to have migrated with purpose to areas they had scouted out before, and it may be that the initiation into cattle-raiding required that cohort to live on the move together for a few years, allowing them to scout out new territory.
The pre-Yamnaya people may have lived in the river valleys for a thousand years before all the pieces came together and they conquered a large chunk of Eurasia in the space of about five hundred years. An ancestral population made up of the same mix of Eastern Hunter Gatherers, Anatolian Farmers, and Caucasian Hunter Gatherers was there for centuries. Intriguingly, the signature y-haplogroup, R1b, is not detected. So what soon became the dominant elite in that group was not yet present - and we don't know where it came from. Were they smarter? Meaner? Males better able to bond with each other and cooperate? Better with horses? Was it a single family that took over fast, Genghis Khan style? Whatever it was, they left enough of a signature that all of Europe, West and Central Asia, and Northern India are descended from them (even if you don't have R1b haplotype). Disease is always a possible explanation, and plague has been detected; constant warfare is another, and that also seems to be the case. Likely, all these things are happening together. If large swaths of population die from disease, others move in to take their land. and conflict can reduce everyone's numbers. So an R1b population with greater plague immunity comes in and its males establish dominance over everyone.
Another interesting bit. The Indo-European languages have long been divided into centum and satem, so it was known that the Balto-Slavic tongues were related to the Indo-Iranian, all the way to Sanskrit, Hindi, and Urdu in Northern India. But once the genetics started coming in, the genetic similarity between the Baltic sites and India seemed impossibly close. There is a second Baltic mystery, of the Yamnaya conquering the Corded Ware people, or so it seemed. That signature R1b replaced the previous R1a. Those might sound like close lineages, but they had actually diverged thousands of years before. Rapid replacement is best explained by conquest, with the victors killing all the males of the vanquished. Yet the Corded Ware people now turn out to be 75% Yamnaya themselves and have pushed back eastward across the steppe eventually making it all the way to India. So who replaced whom? Both. Migration is not with long-term intent. Because we draw arrows on maps it looks as if "those people kept moving westward as part of some plan." But it wasn't a plan. Each year was it's own year and they were only trying to survive until the next one. Large groups of folks move back and forth along migration routes, or to the left and right.
Whether they were intensely patriarchal is also being re-evaluated. Nearly all societies were patriarchal, though there were differences of degree. Marija Gimbutas put forth the idea that the Kurgan People overwhelmed a more gentle, matriarchal Old Europe that worshiped female gods. While other anthropologists thought she oversold this, the idea persisted. The Yamnaya had wiped out everything in their path, it seemed, and that just went together in people's minds. We learned over the last few years that Old Europe was pretty warlike itself, and greater concentration of females in the burials as one moves west suggest something else was up. It's still only 20-30% female, but more than expected.