The question is
how did this movement (Vegetarianism) spread to Hinduism?
Ramayana says (Kishkinda Kanda)
पंच पंच नखा भक्ष्या ब्रह्म क्षत्रेण राघव | शल्यकः श्वाविधो गोधा शशः
कूर्मः च पंचमः || १-१७-३९
"Raghava, five kinds of five-nailed animals,
viz., a kind of wild rodent, a kind of wild-boar, a kind of lizard, a
hare and fifthly the turtle are edible for Brahmans and Kshatriya-s."
Any poet describes/attributes the prevalent customs of his/her period to the characters.
So by the time Ramayana was composed, even brAhmanAs, apart from Kshatriya-s, used to partake meat. Why Vegetarian habits were acquired by brAhmanAs at a later date, is a different issue.
The story of Sage Agastya partaking meat of ram disguised vAtApi during obsequial ceremonies, confirms this.
This Wiki article describes the origins of Jainism.
The Jains claim their religion to be eternal, and consider
Rishabhanatha to be the founder in the present time cycle, the first
of 24 Jain tirthankaras in Jain belief, and someone who lived for
8,400,000 purva years.
The 23rd tirthankara, Parshvanatha, is generally accepted to be based
on an ancient historic human being of uncertain dates, possibly the
eighth to sixth century BCE.
Mahāvīra and Buddha are generally accepted as contemporaries (circa 5th century BCE)
As Ramayana was composed after the commencement of Classical Sanskrit, we can infer that both Vegetarianism and Non-Vegetarianism were prevalent at that point of time.
Even Ramayana describes ascetics/saints Vaikahanasa-s, Vaalakhilyaa-s and those living on leaves, etc.
वैखानसा वालखिल्याः संप्रक्षाला मरीचिपाः | अश्म कुट्टाः च बहवः पत्र
आहाराः च तापसाः || ३-६-२
The sages called Vaikahanasa-s, [who are born out of the nails of
Prajaapati, the first ruler of mankind,] also Vaalakhilyaa-s, [those
born from His hair,] and those from the water of His feet-wash, and
those that thrive on drinking rays of sun and moon alone, and
those that pound with stones and others who thrive on leaves
alone, are those sages...
The Mahayana schools generally recommend a vegetarian diet; according to some sutras the Buddha himself insisted that his followers should not eat the flesh of any sentient being. According to Theravada, the Buddha allowed his monks to eat pork, chicken and fish if the monk was aware that the animal was not killed on their behalf.
So we safely infer that Vegetarianism and Non-Vegetarianism ran parallel in this country since long.