Short answers, pulled from the story.
The Indo-European migrations are hypothesized population movements of speakers of Proto-Indo-European and its descendant languages, taking place from around 4000 to 1000 BCE. They are thought to explain how related languages came to be spoken across a vast area of Eurasia, from the Indian subcontinent and Iranian plateau to Atlantic Europe.
The Kurgan hypothesis, formulated by Marija Gimbutas in the 1950s and later refined by David Anthony as the revised Steppe hypothesis, places the homeland in the Pontic steppe between the Dniepr in Ukraine and the Ural river in Russia, during the 4th and 5th millennia BCE. This region sits north of the Black Sea, Caucasus Mountains, and Caspian Sea.
Sir William Jones described systematic correspondences between Sanskrit, Ancient Greek, and Latin in his Third Anniversary Discourse to the Asiatic Society in 1786, concluding that all these languages originated from the same source. The Dutch scholar Marcus Zuerius van Boxhorn (1612-1653) had earlier noted similarities between European languages, Sanskrit, and Persian, but Jones's formulation gave the idea rigorous footing.
Almost 3 billion people are native speakers of Indo-European languages, making it by far the largest language family. Approximately 60% of all Indo-European language speakers globally speak an Indo-Iranian language, making that sub-branch the largest by number of speakers.
The Yamnaya culture, dated roughly 3300-2500 BCE on the Pontic-Caspian steppe, is considered the source of the bulk of the Indo-European language dispersal. Research by Haak et al. (2015) found that roughly 75% of the ancestry of Corded Ware-related people came from Yamnaya-related populations, and genetic studies have linked Yamnaya migrations to dramatic population replacements across Europe.
The Tocharian languages are known from manuscripts dated to the 6th to 8th centuries CE, found in oasis city-states on the northern edge of the Tarim Basin in what is now Xinjiang, China. They are thought to descend from the Afanasevo culture of Siberia, which itself originated with a migration from the pre-Yamnaya Repin culture at the Don river around 3300-2500 BCE.