There will probably always be crossover from Asian films that break to some extent into western cinema. Tends to be the big epics like Crouching Tiger, Hidden Dragon, Hero, Red Cliff etc.
With western movies themselves it seems the martial arts that are popular themselves have moulded what we see in films. The Bourne movies, Jason Statham films. I also think that any western movie that now has highly choreographed martial arts is going to inevitably be compared to The Matrix fight scenes which probably puts off a lot of film makers.