Both England and France's populations are a mix of several different ethnic groups, going back at least 2000 years.
In addition, non-paternity and mis-attributed paternity rates have been found to be from 2 to 10%, which would cast a doubt over church and states records.
People emigrated from Europe to North America because of the miserable conditions under which they lived. Think about Scotland 150 years ago. Swedish people told me they emigrated to Canada because they were always hungry in Sweden. That wold have been less than a hundred years ago.