Possible follow up: what happened diplomatically between the US and a China as a direct result of this increase in immigration?

It wasn't emigration. It was migration. The majority of people who were going to places like California were doing it as a short term thing. Few people had the intention to stay long term. It was also predominantly men, in part because if was mostly men interested in going, and in part because the US made it more difficult for women to go.

The Six Companies, an association of Chinese welfare associations, acted as an unofficial embassy/consulate for people at the time in San Francisco. Other welfare associations had similar functions in other cities with substantial places. To the extent that the Qing government was involved, they worked with these associations. Mostly though, they had more important things to worry about, and since the 19th century was really quite rough in the south and most of the money earned in the US went back to the south, there wouldn't have been much reason for people in China to be concerned.

The bigger migrations were not to the US but to places like Southeast Asia.

edit: See my replies to the comments here for some more info. If you have specific questions I'll be happy to get to them. Otherwise I've written a fair bit about this period in America and China's history:

• When and went why did Asian Americans go from "yellow peril" to "model minority" in the minds of white Americans?
• On the United States' Chinese Exclusion Act and related anti-Chinese immigration policies
• Why were the majority of immigrants to the new world in the late 1800s and early 1900s primarily from Europe, and not places like China?
• Conditions for Chinese railway workers in North America

I'm happy to go into detail on any other points. Sorry for not having a huge amount of time to put into this just this moment, but please let me know if you have specific follow ups.