Follow-up: Do we know why Elvis learned Karate, as it wasn't extremely prevalent in America yet? Did he make it any more popular in America when he started doing it?

A proper answer would need to consider which martial arts you have in mind. I'll assume you meant Chinese and Japanese, but there are a few others that are rather popular too, like Muay Thai and Filipino Martial Arts.

I can speak for the Chinese part of the question.

The Chinese government has a long-term policy of promoting Chinese culture overseas, and that includes traditional Chinese arts, in particular, their martial arts. This is valid for all its major styles, but let's look just at the example of Taijiquan:

Beginning in the 1950s, a special sports and physical health comission was tasked into making Taiji a simpler, shorter and easier-to-learn activity, expecting that by doing so a greater number of people would practice it.

Because of this, the incredibly long and complicated ancient routines (some lasting more than one hour from start to finish) were condensed into the forms that we see today in parks and academies: 24-movements form, 18 form, 16 form, 42 form, and so on.

This was a very successful strategy, and it's certainly a major factor contributing to popularity of Chinese martial arts in the West. That same method was also used in Wushu (Kung Fu). It's really difficult to find someone willing to learn and retain a 108-movement form, while anyone, even the elderly, can practice a 16-movement and reap health benefits from it.

In all these cases there was also an effort to create new, standarized routines, which allowed Western-style competitions between different schools and countries (i.e. with referees an strict rules, plus flashy clothes and music). This, on its turn, also helped making Chinese martial arts more visible and enjoyable for the Western public. Of course, Bruce Lee and David Carradine also helped, but only because of the systems mentioned above that were already in place.

I believe Japanese martial arts followed a similar path, but I suppose that would depend which country you're talking from. They are very prevalent in Brazil (and South America in general), probably because of its large Japanese population (second only to Japan itself). Brazilian Jiu Jitsu is an obvious mention, but there's also a strong presence of Kenjutsu and Aikido there.

Again, it all depends where you are, but I don't hear that much about Muay Thai and other Far East arts here, although I know they're common in some other countries. I'm not sure if I'd say they're prevalent everywhere.