how to explain "set in toward (a place)" ?

[mengzhu]

eg. The tide of migration that set in toward the shores of North America during the early years of the seventeenth century

I asked some people in some chat rooms,they are from UK or US,they have no ideas..But,however,that sentence is just the 1st sentence of an american middle school history book

[mongoose7]

"that set in" - became established
"toward" - toward.

The tide of migration to North America that became established (became common) in the early 1600s.

(I'm not US.)