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It is said on multiple occasions that initially Hinduism was not a vegetarian society. Was Lord Rama a non-vegetarian? 

I know meat-eating is not restricted to any Hindu Caste, while in general South Indian Brahmins practice vegetarianism. Bengali Brahmins eat fish, and Kashmiri Brahmins eat meat.

and this

There are many references of Sri Rama having meat in the Valmiki Ramayana. It was a common practice in the Kshatriya clan then. Even not many Kshatriyas have a non-vegetarian diet.

It is said that Jainism and up to some extent Buddhism brought the vegetarian movement to Hinduism. As far as I know, Vaishnavism is also responsible for spreading vegetarianism, so my question is how did this movement spread to Hinduism?