So, you think their movement was from Asia toward Europe?
I know I'm really going out on a limb here, but my feeling is that they entered into known, or semi-known, history from far Western Asia. Where they came from before that I haven't a clue.
I think, like you, that the peoples fleeing the Black Sea basin have likely played a major part in history, but I'm not so sure that they were Indo-European speakers.
I took an intense, year-long course in the Chinese-Mandarin language at the DLI. One of the instructors was the leading Chinese Calligraphy expert in the USA. One of his specialties was the history of Chinese writing, and the Chinese people. He had degrees from what he described as the leading university in Beijing before Mao took over.
He described an ancient Han Chinese tradition that the Han tribes had been moving in an easterly direction for a very long time.
My feeling is that Indo-European speakers and Mandarin Chinese speakers have a lot in common, one moving east and the other moving west.