How did the Islamic migration to Europe in modern times come about?
The movement of Muslims to Europe has happened for the same reason most migrations do--economics. Muslims have come to European countries such as England, France and Germany because they feel that there are more opportunities in those countries.
While the vast majority of the migration has had the same cause, there are some differences. Migration to places like England and France has typically come from Muslim countries that were once colonies of the European country. For this reason, England has many Pakistanis and France has many North Africans. In Germany, by contrast, there are many Turks even though Turkey was not a German colony. Instead, they started coming to Germany when Germany created a guest worker program in the 1960s.