I highly recommend you take a look at Coll Thrush's Indigenous London: Native Travelers at the Heart of Empire (Yale UP, 2016). Thrush argues that London should be understood as a place deeply shaped by the many Indigenous North American, Australian, and Polynesian people who have traveled to, lived in, and interacted with the city since the sixteenth century.

Thrush's book is organized chronologically and is absolutely full of accounts of all kinds of Native visitors. As your question alludes to, they came for many reasons, and they weren't always there of their own wills. Particularly in the sixteenth and seventeenth centuries, Europeans sometime imprisoned and trafficked Native people to places like London. "Exotic" domestic servants were quite in vogue for a time, so some Europeans chose to enslave an Indigenous people for social cachet. Commercial profit could also be a motive, especially with the rising popularity of "human zoos" in the nineteenth century.

But Native people also chose to travel to Europe for a variety of reasons. Just as Europeans were curious about "new" places, many Native folks also wanted to explore the world. Education, business, adventure, restlessness: all these factors motivated Indigenous people to travel to Europe.

As British settler colonies increased their hold on Indigenous lands, some Native people traveled to the center of the empire for political purposes. Naaniibawikwe, for example, was a Christian Ojibwe woman from an important political family who made her second trip to England in 1859 as part of a petition campaign to defend Native rights to land around the Credit River. Queen Emma of Hawaiʻi cultivated a close relationship with Britain's Queen Victoria (close to the degree that Victoria was the godmother of Emma's son) by visiting the UK in 1865-66 to much public curiosity and fanfare. This visit was a chance for Emma to consolidate Hawaiʻi's sovereignty by displaying her regalness and ability to navigate polite London society.