What factor accounted for immigration becoming a global phenomenon during the late nineteenth century?
Likely the biggest factor was the decrease in cost and time it took for travel across the Atlantic as well as other places during this time. But in particular, if you look at the wave of immigration that left Europe headed for the United States in particular during the late nineteenth century, much of it was caused in some part by various problems in Europe and Ireland and elsewhere, but for many of the people who made the crossing it would have been impossible if not for significantly lower costs than in previous times.
There were laws and other political decisions that began to limit immigration by putting in place quotas and other measures not long after this enormous wave came to the US in the late 19th century.
I'd say there were factors on both sides of the Atlantic. Inthe United States, sudden and rapid industrialization during the Gilded Age created a huge demand for labor at the same time as the Western territories were being secured and cheap or free land became available. This was an irresistible magnet to people on in Europe and Asia as they sought to escape poverty and chaos in their own nations.
In Europe, economic issues and political upheaval, especially in Germany and France during that time, led to large numbers of germans, Poles, Russians, Jews and Italians immigrating in the same 30 year time span.