I have written a monograph on this topic, titled Settlers and Surnames.
All data drawn from 1880 IPUMS for the USA:
In order to get a sense of how population migrated between 1820-1880, we compare the 1880 county of residence of all persons 40 years old and older with the birthplace of their parents. The birthplaces of parents shown in the following maps are US states and the most important foreign countries. In 1880 the largest immigrant groups were from Ireland, Germany, and Scandinavia.
There is a remarkable tendency for migration to stay close to the latitude of the source state. The resulting migration trace typically moves due west, though eventually fanning out.