No doubt Firmino would flop at Barcelona. He would become Suarez' sub under Ernie 100% or at best stuck on the wing. Would be Griezman 2.0. Griezmann would slide in easily with Salah and Mane as well no doubt.
Manager certainly plays a role in a player's adaptation, but the system and natural fit with teammates is more important.
Firmino would flop here under any manager as long as Messi continues to play his current free role. Wouldn't have what it takes to cut it as a 9 either. He plays with two goal-scoring, pacey wingers at Liverpool. Not two old grandpas clogging the middle with minimal work rate.