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 ...
What is influence of Jainism and Buddhism in introducing Vegetarianism to Hinduism?
Vegetarianism was already a part of Hinduism, and Jainism and Buddhism are religions founded based on Hinduism.
The fact is, according to Hinduism, meat can be eaten only if the animal is sacrificed in a Vedic yajna:
5.31 - ‘The eating of meat for sacrifices’—this is ...