Conversation so far:
What influenced Bring Me the Horizon's music style?
Among Bring Me the Horizon's earliest influences were bands like At the Gates, Carcass, Pantera, Metallica, The Dillinger Escape Plan.
Did they ever make a statement regarding this influence?
CANNOTANSWER.
Who did they model their style after?
Every Time I Die, Norma Jean, Skycamefalling and Poison the Well.
How do they imitate these style makers?
CANNOTANSWER.
Have they always had the same musical style and influences?
However, as their sound developed, the band started to take influences from progressive rock, post-rock, dubstep and electronica.

Question: Did they stick with any one genre?
Self-contained rewrite: Did Bring Me the Horizon stick with any one genre?