How and why did trans-Atlantic and trans-Pacific migration to the United States change between 1850 through 1920?
Immigration across both oceans generally increased in the time period that you are referring to. These changes happened for economic and political reasons. Immigration across the Atlantic also changed in terms of the countries from which the immigrants came.
After 1850, and more importantly, after the Civil War, immigration started to increase. Economically, there was a great “pull” as the US expanded. As the US “opened” the West, huge amounts of land became available. This land, advertised widely in Europe by shipping companies and railroad lines, attracted large numbers of immigrants. Many of these immigrants were from Northern Europe. At the same time, America was industrializing. This attracted many immigrants to the cities to work in factories. Finally, the building of the railroads from the Pacific coast attracted many immigrants from Asia. These were mostly Chinese immigrants who were brought to do the hard and dangerous work of building the railroads through the mountainous West.
There were also “push” factors that caused immigrants to come across the Atlantic. One of the major factors was the increasing persecution of Jews in Eastern Europe. Many Jews came from this region to escape anti-Semitic violence. There were also many people coming from Southern Europe (Italy and Greece in particular). They were leaving partly because the economies of those countries were weak and partly to escape political upheaval.
During the late 1800s and early 1900s, the origins of immigrants changed to some degree. In the Atlantic, immigration from Northern and Western Europe decreased. Instead, immigrants came from Eastern and Southern Europe. In the Pacific, immigration declined dramatically. This was the result of anti-Asian sentiment, manifested most clearly in the Chinese Exclusion Act of 1882.
Thus, immigration across both oceans changed in various ways during the time period referred to in your question.