I haven't been able to find a response to this here, but I apologize if this is a repeat. I'm a teacher who is looking for a more transnational perspective to share with my class than what I was taught and I'm not sure how to answer students who will ask this question. Thanks for any help you can give me.
https://youtube.com/watch?v=RRhjqqe750A
This is an abridged answer, as I'm going to bed, but I'll try to give you one to the best of my ability:
A lot of the immigration from Western Europe was caused by the Potato Famine (it wasn't only Ireland) or the Revolutions of 1848. Once the waves of Irish, liberal Germans, and other minorities in Western Europe died down, it was replaced by other oppressed groups, like the Jews, and Slavic minorities in Austria-Hungary.
A lot of the immigration here was due to increasing intolerance in Russia; anti-Semitism was starting to grow, along with mistreatment of peoples in territory they conquered (one of the reasons why a whole lot of Poles and Ukrainians left for the American Midwest).
https://m.youtube.com/watch?v=nxmWfbXS4Pw
For East Asia, it can be chalked up to two really big things: Qing incompetence, and the constant rebellions. The Taiping Rebellion is notorious for both its length and death toll, and people, seeing an opportunity, started to leave for new shores. Although there was a fair amount of Chinese immigrants coming during the 1849 Gold Rush, a whole lot more started coming, after the fiasco of the Boxer Rebellion. Unlike earlier times of turmoil, Chinese were now able to leave their circumstances, and find new pastures. This was new, as beforehand, they could only go west or south to expand (there was the big fat Pacific, the Japanese, and the Russians to deal with). Now, they could escape.