How has the nature of Hindu practice and belief been affected since the late nineteenth century by being transplanted outside India?
I think that there are a couple of elements at play in such a large question. The first would be that there has been a fairly significant increase in the last century of moving the religion beyond the Indian borders. There has been a broadening of Hindu thought and practice that has helped to increase the Hindu community across the world. Thinkers like Ram Mohan Roy, Sri Aurobindo, Swami Vivekananda, Swami Bhaktivedanta, and, of course, Mahatma Gandhi have enhanced the reach of the religion across the world. Hinduism has become a force that has embodied the essence of "globalization." With this in mind, the religion has become more of an embracing force, something that has been able to be understood across the world and around the world. The establishment of temples all over the world is a part of this. Interestingly enough, I tend to think that the internet has contributed to a widening of belief and practice. File sharing sites like YouTube as well as ISKCON have been able to develop an internet home to Hindu practice and philosophical thought. Being transplanted all over the world and even on the web has enabled the practice of Hinduism to be something that is not limited by geography and border partitions. It is a force of spiritual identity that has demonstrated itself flexible enough to call any realm home.
I think that one way in which the practicing nature of Hinduism has been changed has been to expand its already flexible notion of the spiritual good. Hinduism is a fairly pliable religion in that there are multiple interpretations to grasping the spiritual notion of the good. For example, Vaishnavism, Shaivism, and Shaktism provides flexibility to the religion's definition. A religion that has developed a wide open interpretation to spiritual faith has been even more expanded with its outreach to the West, into a world where borders have become meaningless.
The removal of borders and boundaries has helped to transform the nature of practice and belief in Hinduism. This has been enhanced through the wide acceptance of Hinduism outside India. Organizations that have been dedicated to enhancing the embrace of Hinduism have been critical in this. They have sought to combine both the traditional notions of the practice as well as a modern flexibility in seeking to broaden its borders can be seen in endeavors such as The International Society for Krishna Consciousness (ISKCON). The Hare Krishna movement in America and Europe have helped to broaden the reach of Hinduism. It has impacted the nature of Hindu practice and belief by adding elements of Western worship to Eastern practice. For many Westerners who, for whatever reason might not have felt that their indigenous religion "called" to them, the embrace of Hinduism through organizations like ISKCON has transformed Hinduism in emphasizing its sanctuary element. When Lord Krishna says that any devotee can reach him if they call to him with their heart, it is reflective of how these organizations have tested this maxim, helping to make the element of refuge an intrinsic part of the Hindu faith.