I've provided a brief summary of the decline of Buddhism in India in this previous answer, which contains links to other threads and discussions that go into further detail about the matter.

Regarding Buddhism's spread to east Asia, I can offer additional summary to a very long and complex history for China and Japan specifically. I have less authority on the Korean peninsula, and I have only recently been reading about Buddhism in Tibet and Mongolia. Perhaps our resident Tibet expert /u/jimedorje can provide some insight.

Buddhism appeared in China around the second half of the Han Dynasty during the first century CE. China had a rich and established native religious tradition, with Confucianism being the state ideology of the Han. Other aspects such as ancestral devotion, veneration of rain gods, and worship of heaven were common characteristics. Due to these reasons, Buddhism took a couple centuries before gaining significant ground through China. How it became ingrained to Chinese society was a result of a couple reasons.

Firstly, Buddhism addresses the nature of suffering. Buddhism outlines how to live a proper life in order to end the cycle of rebirth, a repeating process filled with pain and agony. With the fall of the Han Dynasty, China lost a strong, unified, and centralized state structure, leading to hundreds of years of fractured minor states and dynasties. Although major powers would emerge and control parts of China, it would not be until the Tang that a stable state reunified the entirety of China proper. During this period of division, from the Six Dynasties and Northern and Southern Dynasties, many petty kingdoms and states warred with one another. This constant strife naturally brought calamity upon the people, and the notions of salvation promised by Buddhism became appealing. Because of the internal division throughout the realm, many periphery groups, "barbarians", entered the fray and wrestled control over Chinese lands. Among them were powerful steppe peoples to China's north, usually pastoral nomads. As a more mobile non-sedentary people, these groups were in direct contact with the silk road societies and neighbors to China's west. Buddhism since its inception in Northern India had spread across the southern continent towards Gandhara, flourishing there and further moving north into Central Asia and east via merchants.

This leads to the second major factor, with Buddhism as a foreign religion having utility for foreign groups in China. Because many nomadic groups were able to conquer territories of China, they needed to adapt culturally and socially. Notable are the Northern Liang and Northern Wei Dynasties, which had Xiongnu and Xianbei sovereigns control the Hexi corridor and northern plains of China. As foreign rulers, they ruled over the Chinese who already had a strong identity of religious customs, governmental structure, and rich cultural traditions. In many ways China was the center of civilization in East Asia, serving as a model for its neighbors, especially Vietnam, Korea, and Japan in their formation of their states and governments. While the Xiongnu and Xianbei no doubt had strong cultural identities of their own, in an effort to assimilate and rule a new land, they needed to adopt the ways of the native inhabitants, while still counterbalancing with foreign elements. Buddhism offered such a counter, it was a sophisticated foreign religion that paired comfortably with the foreign rulers. Its development from India to Central Asia through hundreds of years had created a rich ideology that matched the advanced religions and beliefs of the local Chinese. This allowed the Xiongnu and Xianbei to validate their claim to authority, by showing the grandeur, sophistication, and even superiority of "barbarian" ways over the native order. State sponsorship allowed for the flourishing of rich Buddhist sculptural art during this time. Buddhism's role in funeral practices and erection of steles became common, with community devotional groups spreading across the realm.

A third factor is the fervent work of missionaries and translators that worked to penetrate Chinese society. Figures such as Kumarajiva translated countless sutras and texts into Chinese, creating the framework for the East Asian Canon that would characterize a major portion of surviving Mahayana Buddhism. Without going too deeply into philosophy, Buddhism had certain parallels with Daoism, Mohism and other Chinese philosophies. Ideas such as universal compassion, self reflection, meditation, austerity, and more were compared. This also aided in Buddhism's spread by reusing native vocabulary and language for new concepts. Buddhism was a major counter to Confucianism, with the established traditions of ancestor worship, filial piety, service to family, society, and state, etc. Buddhist monks with their celibacy, shaved heads, renunciation of family, and independence from state, were strongly criticized and critiqued. However, with time, Buddhism was gradually accepted. One notable example if Huiyuan's treatise On Why Monks Do Not Bow Down Before Kings, which explained how both the layperson and the monastic are compatible with Chinese society by maintaining reverence, respect, and service to social needs. In essence, Buddhism was a very vigorous religion that actively adapted itself to new surroundings with capable advocates and propogators. By the Tang Dynasty, the Han had reassumed control over China, and this marked a golden period of true cultural extravagance and development, especially for Buddhism. The Tiantai and Huayan schools flourished, rich traditions with complex philosophy that explained the nature of the universe and existence. From it Chan and Pure Land would later develop.

From the Tang Dynasty, China could reassert its influence over its neighbors, and it was during this time that Korea would unify under the Goryeo, and Japan would be undertaking its major statebuilding efforts in the Nara court. China, as a strong civilization with unparalleled refinement in art, music, crafts, religion, government, technology, etc, were modeled after keenly. Through this, Buddhism would spread east to Korea and Japan. As stated above, I am more well read on the literature for Japanese Buddhism, so I'll go into a bit more detail here. Buddhism in Japan initially began as a religion reserved for the elite. While Japan had its native Shinto belief system, Buddhism was seen as a mark of sophistication and development, only appropriate for the nobility. The study of Buddhism was very academic, with the philosophy being a more scholarly subject than a personal emotional pursuit. Rituals were held to bless and protect the state, call for good weather, or heal sick persons. Buddhism was an extension of the Imperial Court's authority, and priests would come from noble backgrounds, with temples built across the islands as a visible entity of state control. Later on the developments of Shingon and Tendai allowed for greater exposure of Buddhist doctrines, as well as dispersal towards the common people. Zen and Jodo would also develop later on in correspondence to its Chinese roots, and native Japanese ideas and schools also emerged such as Nichiren. Although Shinto existed as an established religion, Buddhism did not directly interfere with it. Native deities were appropriated into a Buddhist context, and Buddhist rituals and customs existed alongside reverence of indigenous kami. Jumping forward to the Tokugawa regime in the 17th century, isolationist/insular policies further ingrained Buddhism into Japanese society. Christian rebellions were brutally suppressed, and the implementation of a temple registration system, the danka, saw the mandatory submission of Japanese families to local temples.

Similarly, Christianity was seen as a dangerous religion for national security in Korea and China, and was suppressed accordingly, though Korea in the Joseon Dynasty was a very Confucian society, and Buddhism had began to decline in China after the Tang. While the golden age of the Tang had passed, Buddhism's richness and advancement in philosophy influenced the development of Daosim and Neo-Confucian thought, while also existing on its own as a major religion. The modern period has its own complex history of upheaval and massive social and cultural shifts. Imperialism, revolution, and conflict has seen China, Korea, and Japan change drastically. New religious groups flourished in 20th century Japan, Christian missionary work in South Korea saw tremendous success, and Communist governments in China, North Korea, and Vietnam have seen suppression and liberalization of religious expression. Chinese Buddhism is experiencing a major revival on the mainland, and is maintained extremely well in Taiwan.

Buddhism saw significant challenges early on in China against established customs and religions. However, as it gained its position in China, it expanded beyond when other countries were in close contact with Chinese civilization.