The mixture between Iranian & Pakistani people refers to Indo-Iranian people. We are not referring to Persian mixture in the Pakistani populace because that would naturally be less. The Pashtuns, Balochis, northern Punjabis, & Kashmiris are all descendants of the Indo-Iranian migrations. Those migrations took place from Central Asia or Andronovo, it's important to remember that ancient Central Asia was different from the modern ethnic groups residing there. Pakistanis are not close to Turks at all, no genetic study ever indicated that. In fact all studies indicate relations to Indo-Iranian people & to an extent Eastern European people. Some Turkic groups were residing in Central Asia during the time the Indo-Iranian Sintashta culture was prevalent in Central Asia, but there is no known admixture between them.
Any Semitic or Turkic mixture present in the modern day Pakistani population took place within the Islamic era. That mixture is so little & in so few people that we might as well not consider it or even discuss it. The Sumerian term Meluhha was applied to the Harappans, but that does not imply that the Harappans migrated there. In fact, the Harappans are only known to have business relations as in trade with the Babylonians & Mesopotamians. Languages do not always travel through migration, they also travel through cultural or political dominance. However, in the case of the Sub-Continent, Afghanistan, & Iran, Indo-Iranian migrations took place towards Iran, Afghanistan, & the Indus Valley. Once the Indo-Iranian people settled in the north western regions of the Sub-Continent, their cultural dominance spread their languages all over the region. This is why Indo-Aryan languages are spoken all over the Sub-Continent even though most people do not descend from them. Genetic studies confirm this, & the reason the caste system was developed was to preserve their power & race. The Indo-Iranian people remained predominantly in the north western regions of the Sub-Continent.
Those dates (particularly the 1500 BC migration) are not accurate because they are generally derived from Max Mueller's Aryan Invasion Theory which has been discredited. Max Mueller derived those dates in a manner that got them to conform with his religious beliefs in the Bible & its dates of the world's creation or something. That is why those dates are inaccurate.
The Indo-Iranians or Aryans migrated towards the Indus, they did not invade or enslave it. Their migration is said to have occurred around 3000 to 8000 years ago. The source that you asked for is present below.
Genetic evidence suggests European migrants may have influenced
the origins of India's caste system
This study focuses on India, but the dates for Indo-European migrations predominantly towards the north western regions of the Sub-Continent are still valid.
Please note that Sanskrit was at one point in time an unwritten language. In fact, I have read that the Aryans developed their script after interacting with the Harappans & other local people of the Sub-Continent.
That's not correct, genetic & historical evidence does not indicate that at all. The Vedic Aryans considered themselves distinct from the Harappans, even though they did marry their women sometimes. They migrated towards the Indus Valley in the later stages of that civilization & probably resided in Afghanistan before that. The Sanskrit language is different from the Harappan language. It isn't reasonable to believe that those people randomly forgot their language or history after the IVC's collapse. The term Aryan evolved in a manner similar to the word Roman. In the later era of the Roman empire, the Byzantines called themselves Romans even if they were Greek. That doesn't change the fact that the Romans are the people of Italy. Similarly, the term Aryan evolved in to meaning anyone that follows the Vedic culture or speaks their languages. It initially referred to a race, similar to Avestan's word "Arian". There is little proof that the people of the IVC were monotheists, & we won't know much till we decipher their language. The Vedic Aryans were not monotheists though.
Deciphering their language will help us to classify it as Indo-European, Dravidian, or even the possibility of it being Semitic, even though that is extremely unlikely. In the past, the original speakers of a language shared heritage with the original speakers of their language's sister languages. We already know that the Harappans hadn't been colonized previously so there language is bound to be their own. No other civilization has a script similar to theirs.
Anyway, this was a nice topic for discussion, but I am bored of it now. Let me know if you want me to provide you with more links of genetic studies or some other relevant sources.