Here is the fun part. Everyone thinks Pranayama and Yoga are from Buddhism. But really it's from Hinduism (it is mentioned in Vedic scriptures). Buddhism adopted these techniques which invariably resulted in cultural appropriation.
However, it doesn't matter what the source is as long as the technique is useful for the end user.
And even more fun, Yoga share a lot of philosophical roots with Samkhya, a school of Hinduism that doesn't exist anymore. Samkhya came about at the same time as Buddhism and is also atheist, or at least doesn't concern itself with god, much like Buddhism.
Yoga, however, brought god back into the mix, and these days, when taught in India, it's mostly done in the tradition of Advaita Vedanta school, which is the vast majority of hindus.
Advaita means "indivisible", i.e. god, the world, and everything is one thing, but the very few texts of yoga that has survived, clearly show it was a dualist philosophy separating prakrti (matter) and purusha (consciousness).
It's not just taught in Advaita schools. It's also taught in Dvaita and Vishisht-Advaita Vedanta schools. I know because I belong to the Dvaita school of thought. Also, there is a big misconception that majority of Hindus are Advaitis. It's not so. Almost half of Hindus are Vaishnavas who either belong to Dvaita, Vishishtadvaita or Dvaita-Vishistadvaita Schools (a.k.a ISKON/Chaitanya Mahaprabhu).
Yoga itself however originated from Santana Dharma (or modern day Hinduism). To back up this statement, you can read this chapter of Mahabharata which talks about the birth of Ved Vyasa (the writer of the Vedas) wherein it mentions how the Sankhya, Yoga, Pancha-ratra, Vedas and Pasupati cults came into existence: http://sacred-texts.com/hin/m12/m12c049.htm
I'll quote the relevant text for your convenience:
"Vaisampayana continued, "I have thus narrated to thee, O monarch, the circumstances connected with the former birth of our revered preceptor, viz., Vyasa of unstained mind, as asked by thee. Listen to me once again. There are diverse kinds of cults, O royal sage, that go by diverse names such as Sankhya, Yoga, the Pancha-ratra, Vedas, and Pasupati. The promulgator of Sankhya cult is said to be the great Rishi Kapila. The primeval Hiranyagarbha, and none else, is the promulgator of the Yoga system. The Rishi Apantaratamas is said to be the preceptor of the Vedas, some call that Rishi by the name of Prachina-garbha. The cult known by the name of Pasupata was promulgated by the Lord of Uma, that master of all creatures, viz., the cheerful Siva, otherwise known by the name of Sreekantha, the son of Brahma. The illustrious Narayana is himself the promulgator of the cult, in its entirety, contained in the Pancharatra scriptures."
Hinduism eats other religion, it's the super set religion, which is cool except for the cast system and that it says we all are living in the downward spiral (Kali Yuga) where people are as far away from God as possible. I don't wanna live in a downfall...
The caste system is really not prevalent within Hinduism. Again another hit job by the British (& most Western media) to discredit Hinduism and paint it in a bad light. In fact the word "caste" is Portuguese in origin, there is nothing equivalent to caste in Hinduism.
Also, I'm wondering how you got the impression that Kali Yuga = downward spiral. It is a phase in the endless cycle of the universe, but not necessarily a downfall.
However, it doesn't matter what the source is as long as the technique is useful for the end user.