Maybe the Mongols chase them over?
The primary cause of the conquest of the New World was disease. This works for any immigrant group that includes children with measles, smallpox, rubella, chickenpox, pertussis, mumps, influenza, etc.
So we have middle age villages inside wooden stockades, with Indian raids as their population recovers. Iron, copper, and lead smelting, some byproduct silver, some gold in North Carolina and Virginia, maybe some salt mining eventually, lime and gypsum kilns, glass and ceramics, textiles, etc.
Horse barbarians rule the plains. Some Indians and some Europeans and some mixed. The East involves mostly Europeans. No Africans without an organized slave trade. Some Moslems brought over as slaves but they look too much like Europeans to stay inslaved because the small and independent states make it easy to run away.