Go became "international" 200 years before the earliest forerunner of chess had even been invented. That's when it spread from China to Korea. And it reached Japan around the same time chess's earliest forerunner was being invented in India.
Beyond that it seems likely that dice games from e.g. Romans, Egyptians, Assyrians were likely the first to spread as those empires spread and the soldiers who played them played them with locals.
Beyond that it seems likely that dice games from e.g. Romans, Egyptians, Assyrians were likely the first to spread as those empires spread and the soldiers who played them played them with locals.