No that's mostly wrong. A group of Indo-Europeans did enter India, but North Indians were likely relatively lighter-skinned at that point already. It's true that Brahmins (the priestly caste which had the most privileged status in the ritual hierarchy, though they lacked the economic wealth of business-oriented castes) have the most Indo-European steppe ancestry (controlling for region, as North Indian lower castes have more steppe ancestry than South Indian Brahmins), but beyond that the correlation is much more complex. Many tribals and untouchables in North India have more Indo-European ancestry than the wealthy business castes for example.