While Lehi's vision begins with a very personal revelation about his sons, the lesson is clearly expanded to show how the world fits into the pattern. In the vision, Lehi's family becomes a type and model for the rest of the world, both for those who would come to taste the fruit, and for those who would not. The entire world follows this same search, just as individually as did Lehi's family.
The "concourses of people, many of whom were pressing forward" represent the inheritance of spirit we bring with us to this world. It is in the nature of our spirits to want to press forward to obtain this fruit, even if our conscious minds do not recognize that fact.