There is plenty of evidence for the migration of speakers of Indo-European languages from the area that is now the NW quarter of Kazakhstan and Russia between the Kama river and that part of Kazakhstan:

• This is the red area on this map: https://commons.wikimedia.org/wiki/File:Andronovo_culture.png

• Modern borders for comparison: https://commons.wikimedia.org/wiki/File:Caucasus_central_asia_political_map_2000.jpg

From there, they went east, west, and south, taking languages (Indo-European languages), technologies (such as the spoked-wheel chariot), and genes with them, reaching China in the east, Ireland in the west, and India in the south. This Indo-European expansion is well-supported by linguistic evidence, genetic evidence, and archaeological evidence of the spread of technologies. There are many details that are unknown, because much of this was not recorded in writing at the time, and languages, genes, and myths don't tell us the whole picture. But despite the lack of detail, the broad picture is soundly based on clear evidence of multiple types.

The Indo-European peoples who moved south became the Indo-Aryans, speakers of Indo-Iranian languages ("Iran" can be translated as "land of the Aryans"). It is believed that Indo-Aryan peoples, having moved into and through Iran, reached India in about 1500BC, which is also the date usually given for the oldest Vedas. This is the time that Iranian and Indian Indo-European religions began evolving apart from each other. Zoroastrianism did not wipe the slate of Iranian religion clean, but kept a lot of elements from the older polytheistic Iranian religions. Therefore, a great deal of similarity is to be expected between Zoroastrianism and Hinduism. There is also a great deal of similarity in languages, with Avestan (in which the Zoroastrian scriptures (the Avesta) were written) and Vedic Sanskrit (in which the Vedas were written) being closely related (and also the oldest well-known Indo-Iranian languages, due to the Avesta and the Vedas).

The similarities between Hinduism and Zoroastrianism pointed out in the article you link are nothing more than what is to be expected given the common origin of both religions in older Iranian religion, and the evolution of Avestan and Sanskrit from a common ancestor. Given the strong evidence for a movement of Indo-Aryan peoples from Kazakhstan to Iran and through Iran to India, it doesn't appear likely that Zoroastrianism came to Iran from India (or Kashmir).

Given that, why does this article suggest this? The very idea that the roots of Hinduism or Sanskrit are to be found outside India is anathema to many Hindu nationalists and some varieties of Hindu nationalism (most prominently, Hindutva). Instead, it is assumed that the Indo-Aryan peoples, languages, and religions of India were already there, long before the supposed Indo-European migrations. Given the similarities between Hinduism and Zoroastrianism, and Sanskrit and Avestan, claims that language and religion flowed north, from India to Iran, are important for sustaining the idea of Indian origin. That this is contrary to the widely-accepted evidence doesn't appear to matter to those who push these ideas.

Basically, it's revisionist history that ignores a very large body of evidence in order to support nationalist claims. That there are many similarities between Hinduism and Zoroastrianism, and Sanskrit and Avestan, is beyond doubt. What is wrong is the conclusions that the article claims are based on this.

For a look at the kind of passions this topic can inspire, see the discussion about this article: https://parsikhabar.net/culture/ancient-persian-influence-on-hinduism/750/