Well, sort of. Chess spread from India, but in many different forms. The form we call "chess" today is only around 500 years old, although the original game is over 2000 years old. And even today, the Chinese and Japanese have chess games distinct from Western chess (Xiangqi and Shogi respectively). It's interesting that Go didn't similarly "speciate". Perhaps because unlike chess, there wasn't the multitude of pieces and movements that seemed arbitrary.