Rinzai's koan curriculum is the part that travelled. In Japan its history is at least as much about proximity to power, and the two are connected.
The gozan system
From the fourteenth century the Ashikaga shoguns organised the leading Rinzai monasteries of Kyoto and Kamakura into a ranked network, the five mountains, with state-appointed abbots. The monasteries handled diplomacy with China, ran trade missions, and produced a literature in Chinese verse. Rinzai became the school of the capital and of the warrior aristocracy, while Soto grew in the provinces.
What that produced, and cost
The association gave Japan the tea ceremony, dry landscape gardens, ink painting and much of what is now exported as Zen aesthetics. It also tied the school to whoever held power, and the twentieth-century consequence has been examined unsparingly: senior Rinzai figures supported Japanese militarism in explicitly Zen terms, and several institutions have since issued formal apologies for it.
Hakuin's repair
By the eighteenth century the gozan system had decayed and koan practice had become largely formulaic, with answers transmitted rather than reached. Hakuin rebuilt the curriculum from the country temple he refused to leave, insisted on the sustained doubt that a koan is supposed to generate, and taught that the work continues after awakening rather than concluding with it. Every Rinzai lineage in Japan today descends from him, which is a measure of how complete the collapse before him had been.