In the late 1800s, migration happened for many of the same reasons that it generally does.
For example, there was migration motivated by economics. This could be seen, for example, in the migration of Chinese and Japanese to places like California and Hawaii. This was also a major factor behind the migration of Italians to the United States.
There was also migration motivated by fear of religious persecution. A major example of this was the migration of Jews from Eastern Europe. They fled (though this was not the only factor) increasing violence against them in places like Poland.
Finally, there was migration due to political persecution or uhpeaval. This was one of the causes of migration out of Eastern Europe during this time.