One of the major differences between Asian and Western fight scenes, common wisdom has it, is that in Asian cinema the fight scenes continue to tell the story, while in the West the story stops cold until the action is over, and then continues on its merry way.
There is some truth to that - the Burly Brawl in Matrix Reloaded, for instance, adds nothing to the plot, and doesn't even function as a good martial arts scene because it devolves into a CGI-fest as soon as it can. But there are several Asian action films in which the fight scenes also add absolutely nothing to the plot - but are the only reason anyone would even want to watch the movie in the first place.
But there are Western films in which the action or fight scenes are perfectly integrated with the plot and the development of the characters. Case in point: Scaramouche, where Stewart Granger watches his friend get killed by José Ferrer in the first fight scene and is incapable of doing anything about it as he's not trained in fencing at all; faces Ferrer again later on as he's already had training but isn't his equal yet by a longshot, and barely escapes with his life; and finally faces Ferrer in the climactic duel in the theatre which ends the film.
Yet even in pure drama, fight scenes can also tell the story of the film.
In Million Dollar Baby, the first boxing match which opens the film shows us Clint Eastwood in action as the best cut man/trainer/manager in the business.
Later on in act 2, the first match we see of Hilary Swank has her being defeated until Clint steps in and takes over as her manager, and immediately helps her win the match with his advice. This match corresponds with Focuspoint (or Pinch) 1, and shows both Clint accepting responsibility for Hilary and Hilary's first step on the road to boxing success.
The midpoint is the match in London, where she wins her bout and gets the 'Mo Cuishle' robe from Clint. It's the most triumphant moment of the film for both of them, and the robe symbolizes that he has come to see her as his daughter (Mo Cuishle meaning 'my darling, my blood' as we learn in the climax of the film). They're both on top of the world at this point.
The final big fight scene is the championship match against the Blue Bear. It's the most savage fight in the whole film, and we suffer along with Clint as he winces at each brutal blow Hilary has to endure. The stakes are very high, and the whole fight (and act) come to a horrifying end when a sneak attack after a round has ended brings Hilary crashing down onto a foot stool to break her neck. This is the second plot point which catapults the film into its powerful and surprising third act.
If your script has action or fight scenes, see how far you've integrated them with the big structure beats of your story. The more action scenes are inextricably linked with character and plot development, the more emotional impact they will have on your audience.