Here in the west, when we think of "Buddhism" we probably imagine a solemn and dark temple with rows of shaven head monks sitting earnestly in deep meditation while striving with all their might towards enlightenment. In other words our perception of Buddhism is centered on the deeper philosophical and mystical elements of the religion.

While this type of Buddhist practice certainly existed, it was not and is not the normative experience of people who are actually engaged with the religion in its Asian homeland. Depending on the particular country, the average Buddhist center will likely be a priest's family's home, a temple that largely functions as a community center, a place of devotional worship or even a museum-like tourist attraction. What you don't see hardly any of is people meditating.

Because of this dichotomy between what people in the west were interested in and what was happening at 99.9999% of Buddhist centers, you did not and don't see alot of crossover. If someone is going to travel Asia to practice enlightenment-in-this-lifetime Buddhism, they are not going to hang around the local community center with a bunch of grandmas or follow a priest around as he visit's people in the community to chant at memorial services at their homes. Instead they are going to visit centers that specialize in meditation and the general public are generally not involved at those centers.

When people did travel to Asian to practice meditation-based Buddhism it was typically places that explicitly welcomed westerners and these centers were usually happy to have them. If you were (or are) a Buddhist teacher that had deep spiritual aspiration, there is a good chance you would be disappointed in lack of aspiration in many of the locals who were practicing with you. If you look at Japanese Zen for example, there are very few "training monasteries" were meditation is actually practiced. The vast majority of the young men who are enrolled and living at these monasteries are not there for reasons of religious or spiritual aspiration. They are there because they are the first sons of a temple family and they are expected to take over and run the rural temple they grew up at when their father retires. In order to get their priest certificate one of the steps is that they needed to spend one to three years at one of these monasteries. Its more a rite of passage and a feat of endurance than anything else. So when westerners with a burning spiritual aspiration started turning up, some Zen teachers were ecstatic and welcomed them enthusiasm. Generally word would get around to which teachers were enthusiastic about teaching westerners and in many cases those monasteries would end up being primarily populated by westerners.

Of course there would be inevitable cultural clashes and misunderstandings, but generally people were traveling to practice centers that were set up to welcome them and work with them.

One place were there was the type of clash you are asking about was in Zen temples in the United States. Cities with many Japanese immigrants would typically have a Soto Zen Buddhist temple in the area to provide the traditional cultural services they were used to at home. These were primarily culture centers as mentioned above and none of the Japanese congregants had any interest in meditation.

But what started happening in the 1960s as the beat generation and then the Hippies started developing an interest in Zen is that many of these young people would show up at these community center-like temples looking for a Zen Master. In two cases (in San Francisco and Los Angeles) the resident temple priests were interested in working with these young seekers and a meditation practice started developing at these temples. The Japanese congregants would come around for traditional cultural services and the young westerners would come around for meditation. Inevitably this would cause irritation with the Japanese that their cultural centers were being invaded and jealousy that the priests they imported (and whose salary they paid for) were giving so much attention to the westerners . In both the Los Angeles and San Francisco, the priests ended up leaving the Japanese temples and starting Zen Centers populated almost entirely with westerners.