Regarding Barkley's situation under Koeman. Throughout Martinez's tenure, Martinez never strayed from his 4-2-3-1 formation, meaning Barkley was always first choice for the Center attacking role.
The issue is less so Barkley's ability, though his form is in issue. The issue is Koeman's demands. Koeman is pragmatic, and we haven't seen the 4-2-3-1 regularly. And the problem is that Barkley doesn't have a specialist position. He'd be best utilised having free reign of the attacking third, almost like a De Bruyne role. However this isn't Koeman's preferred system.
Barkley's defensive work is really lacking, and in situations where you need a team to defend as a unit like last week he isn't the best option. The reason I've really been impressed with Koeman is that he adapts to games and the opposition not just before a game but during. In a way, this is the complete opposite to Martinez who was unflinching and stubbornly clung to a philosophy. And I'm not saying this isn't an effective approach. Koeman's will change the formation and players how he sees necessary.
There are a few players who will always have a place in the team - Lukaku, Gana, Williams and Stek, maybe extended to Coleman and Baines. In between that spine the players are interchangeable. And therein is the Barkley issue. There are already signs that Koeman isn't shy of stamping his authority, even a player like Holagte who had a superb start to the season gets dropped for Coleman as soon as he's fit.
Other than the spine of the team I mentioned we never really know who will play and where. And this has given Barkley a reality check, because he has been in the mindset that both his position and place on the team is safe regardless. What we are seeing now is his struggle to adapt to these changes. As to what will happen to Barkley, it is still early days in Koeman's management. This is why I have some sympathy for him, this dramatic change has affected him mentally, and now he is in an uncertain period in his career.