All around the world, martial arts from many different countries are a major cultural export. I say this because it’s important to remember that your characters race, ethnicity, and gender don’t necessarily need to reflect their style’s culture or country of origin. One of the greatest beauties of the MAs is that anyone can start at any age and find both meaning and value. I’ve seen twelve year olds earn their black belts side by side with eighty year old cancer survivors. While I trained in Taekwondo and the master instructor of my dojo was Asian, he was not Korean, instead he was Japanese-American. His master and the master co-founders of the entire organization were a Filipino-American man and an African-American man. The instructor who had the greatest effect on me was (or his family was) an immigrant from Ecuador.
A good martial artist can come from anywhere and while they can’t necessarily be just anyone (just those who put in the time and effort), there’s no need for an author to limit their imagination with a student of any traditional art because their race, gender, or ethnicity doesn’t fit with what media has prepared us for.
When it comes to martial arts specifically, it’s important to remember that the make-up of a school you might expect is not there in actuality. Many people are drawn to the martial arts from many different walks of life, and while there is certainly some very interesting mysticism and philosophical tenants grounded in some of the Eastern MAs, in a modern context the techniques and philosophies are more than able to transfer into backstories the author needs. All the author needs to do is be aware and sensitive of the culture and philosophy ingrained in the martial art they choose.
If you are working with a historical context, research as needed. Either way, you might be surprised.