The Brahmajala Sutra, a Mahayana text setting out the bodhisattva precepts, states directly that a bodhisattva must not eat the flesh of any sentient being, reasoning that all beings have at some point in the endless cycle of rebirth been one's own parent or child and that eating flesh is therefore a kind of cannibalism across lifetimes. This textual basis existed before it became binding practice, and for a long stretch of early Chinese Buddhist history monastics, like their Theravada counterparts elsewhere, ate whatever was placed in the alms bowl, meat included.
The turning point was political as much as doctrinal. Emperor Wu of Liang, a devout Buddhist ruler, issued an edict in 511 CE requiring monks and nuns within his realm to abstain from meat and alcohol, citing the Brahmajala precepts as authority and backing the requirement with the power of the state rather than leaving it to individual conscience. The edict took hold and became normative for Chinese Buddhist monasticism from that point forward, and Vietnamese and Korean Mahayana Buddhism inherited the same standard through Chinese transmission.
This produced a durable divide within world Buddhism: Mahayana monastics in East Asia are strictly vegetarian by institutional rule, while Theravada monastics in Southeast Asia and Sri Lanka eat whatever is offered as alms, including meat, so long as the monk has no reason to believe the animal was killed specifically to feed him, the so called rule of the three kinds of purity.