Industrialization, immigration, and urbanization all took place on an unprecedented scale during this era. what were the relationships of these processes to one another? how did each process serve to catalyze and fuel the others?

Industrialization, immigration, and urbanization are all immensely tied into one another. By the end of the 1780's, England was the first country that industrialized itself. Industry reorganized the division of labor from one previously focused on agriculture to that of manufacturing. Industrialization was encouraged because it increased the speed of production and mobility. 

Leading to a primary focus in manufacturing lead to urbanization, or the mass migration to urban areas where manufacturing may be made possible. This influenced large numbers of people to leave agrarian life for life in the city, and with large factories of industry demands large numbers of workers. Industrial countries such as England and the United States gave foreigners the impression that working in a factory or manufacturing will correlate to better work opportunity. This made industrial cities like New York, London, etc. popular destinations for immigrants to find work. 

Why do you have to ask such hard questions.