Ok. There seems to be a bit of compressing the timeline and geography here. It's not entirely accurate to say that anybody derived their culture from Indo-European roots, since that's technically a linguistic term, but there are cultural (mostly religious) elements that followed similar trajectories. You also can't say that the European branches of the family tree derived from Indo-Iranian because Indo-Iranian is another branch off of Proto-Indo-European.

To determine where the shared ancestor of Indo-European languages originated linguists identify shared root-words across languages from Portugal to Bangladesh to see what environment they would have described. That's what suggests that it originated in the Pontic-Caspian steppe of modern Ukraine and eastern Russia. To determine when that shared language - Proto-Indo-European or PIE - was spoken, linguists use a series of predictable and consistent patterns of change in the descendant languages. That produces a date of c.4000-2500 BCE.

The Sintashta Culture is both largely east of that region and 100-700 years after that date (lasting from c. 2400-1700). When we talk about a prehistoric culture like this, it's usually an identification of similar art and tool designs that takes its name from a particularly important archaeological site - in this case a town called Sintashta. However, there is a connection. The dominant material culture in the area that PIE originated at the time that PIE was spoken would have been the Yamnaya Culture, which existed in the area north of the Caspian Sea, Caucasus Mountains, and Black Sea from about 3500-2500 BCE. Yamnaya existed in the right place, in the right time, and represents the first widespread use of kurgan burials; hence the Kurgan Hypothesis in which the spread of Indo-European languages can initially be traced through the spread of kurgan tombs.

Neither linguistics nor genetic studies nor comparison to earlier material cultures give a conclusive explanation for who preceded the Yamnaya, and therfore no explanation for who preceded the PIE speakers, but they all shed a lot of light on what happened to groups that split off.

• Based on linguistics, at least one group of PIE speakers broke off just before the Yamnaya Culture emerged and moved southwest (or an even earlier Proto-Proto-Indo-European language according some theories). This was the progenitor of the Anatolian Language Family, which is now extinct but includes ancient languages like Hittite and Luwian.
• Around 3000 BCE, right in the middle of the Yamnaya period, some ancestors of the Tocharians set off east. Eventually they reached modern Xinjiang, China. This date is confirmed by material culture known as the Afanasievo and a linguistics quirk that I'll come back to later.
• By no later than 2700 BCE, there was significant movement west with both genetic and material cultural influence from the Yamnaya on the Corded Ware and Bell Beaker cultures in Central Europe and vice versa. Linguists debate the exact order that different European Language families split apart. Either Germanic or Hellenic (ie prot-Greek) left first. Then the ancestors of the Italic languages split off (mostly represented by the Romance languages today) , followed by some combination of Albanian, Celtic, and un-categorized ancient languages going west and proto-Armenian going south. Celtic used to be considered an obviously close relative to the Italic languages, but more recent theories suggest it may have been the very last westward breakaway from the end of the Yamnaya/PIE culture.
• The predecessors to the Indo-Iranian and Balto-Slavic languages migrated east by 2500 BCE, at which point we kind of lose track of Balto-Slavic for a few millennia. Most of the languages referenced in eastern Europe and Central Asia by ancient authors are obviously Iranian, including those in the original PIE homeland, but Balto-Slavic languages were clearly spoken somewhere in this eastern region and share important linguistic features with Indo-Iranian. There is a strong genetic connection between Finnish populations and the Yamnaya culture, so it is possible that Balto-Slavic represents some kind of northern migration out of the original Yamnaya/PIE complex.

All of these later eastward movements are contrasted from the earlier migrations by linguists using a relatively simple linguistic difference: the word they use for 100. Every Indo-European branch prior to c. 2500 BCE used a word that started with a hard "K" sound. This includes Greek and Germanic languages like English even though that "K" ultimately shifted to an "H." These are called "centum languages," after the Latin word for 100 (all ancient Latin C's were hard "K" sounds). They are contrasted with "Satem languages," named for the Sanskrit word for 100. In these languages that hard "K" sound transitioned to a soft "S" sound. This phenomenon should be pretty familiar to any English speaker based on our use of the letter "C." That's one of the easiest ways to tell that Tocharian split off from PIE before the larger eastward migrations.

That or a northern movement out of the predecessors to the Sintashta Culture. Around 2700 BCE the so-called Poltavka culture developed as an offshoot of the Yamnaya around the Volga River, but by 2500 a much more developed and widespread form of some of the same material culture had spread through the region north of the Aral Sea. That's the Sintashta. This is the first Central Asian culture with mass use of thing like three-bladed arrow heads, chariots, and copper and bronze metallurgy. All of this was well preserved because of the continued use of kurgan burials. Based on the location and time period, if we assume the Yamnaya culture were speakers of PIE, then the Sintashta Culture would have been speakers of Proto-Indo-Iranian.

By about 2000 BCE, the cultural halmarks of Sintashta were spreading and changing eastward, eventually reaching as far as the Altai Mountains in southwestern Siberia and as far south as the Kopet Dag mountains in Iran and the BMAC culture of Bronze Age Bactria. Of all of these prehistoric material cultural groups, the Andronovo are the most obviously connected to a specific linguistic heritage. Andronovo sites contain many clear antecedents to practices described in the most ancient written sources from Indo-Iranian lanuages: The Vedas from India and Old Avestan from Zoroastrianism. It also conveniently overlaps with the eastern half of the material culture that we identify as Scythian or Scytho-Siberian, but that material hadn't actually developed yet.

The arrival of Andronovo sites/Indo-Iranian speakers coincides pretty precisely with the gradual decline of the urbanized BMAC culture around the Oxus River/Amu Darya around 1900 BCE, but so far as we can actually tell whatever migration was driving the spread of Andronovo material culture didn't actually stop at the Kopet Dag. Instead, Indo-Iranian speakers seem to have just kept on going, but moving into environments that didn't support the same Andronovo-style steppe culture. The split that went southeast, eventually formed the Indo-Aryan or Vedic language group that gave rise to the "Indo" part of Indo-European by 1500 BCE, with both ancient languages like Sanskrit and modern ones like Hindi ultimately arising from that branch. However, a part of the same migration also seems to have gone west. Most famously, the Mittani kingdom in modern Syria was seemingly ruled by a class of expert charioteers that imported Vedic-like names for themselves, their gods, and equipment related to horses as well as some Vedic-like prayers around 1600 BCE. Around the same time, the ruling elite of the Kassites - a culture from the Zagros mountains that conquered Babylon - also had seemingly Indo-Iranian names.

That migration left the remaining members of the Andronovo Culture and Indo-Iranian speakers to develop into the Iranian language family and all of the cultures that would ultimately be tied to that. This includes the languages of historical Bactria and Sogdia, the Zoroastrian scriptures of the Avesta, and the languages spoken in the region of modern Iran. The Iranian languages first began appearing in that region in the early Iron Age, possibly drawn in by a combination of climate change pressures in Central Asia and the economic shifts of the Late Bronze Age collapse (they were hardly the only group on the move at the time).

After about 1000 BCE, the Andronovo Culture began to fade and some new, highly distinct styles of art, tools, religion, and warfare emerged. That's the beginning of the culture - really group of similar cultures - that we know as the Scythians. The emergence of the Scythians came at the very end of this period of pre-historic linguistic and material cultural movements on the Steppe and represent one of the few migrations/cultural spreads that moved back toward the Black Sea.

See especially:

The Horse, The Wheel, and Language by David W. Anthony

The Origin of the Indo-Iranians by Elena E. Kuz'mina