I have a number of family members that migrated hundreds of miles. My Joshua Badger (IV) moved from his first appearance in Wilkes County,North Carolina to Fairfield County, SC to Tennessee to Mercer County, KY to DeKalb County, GA. The distance between these five locations is 786 miles. This is many hundreds of miles farther than the distance you find improbable.
Joshua (III) wife moved from New Jersey to Western, PA and to Clarke County, KY. Again this is many hundreds of miles farther than you are suggesting is not probable.
Please keep in mind that the road systems from Connecticut and Massachusetts to New Jersey were much better than the known wilderness treks of made by Joshua IV. Washington and Rochambeau marched over 10,000 men this distance from Rhode Island to Virginia in a matter of weeks.
Many of my ancestors traversed the wilderness and hunters trails from Pennsylvania to the western counties of Virginia and the Carolinas and then into the Kentucky and Tennessee areas. The distance argument is not valid for ruling out identity. If some one could show that any of the Joshua Badgers were in different places at the same time, then we would have proof that they were not the same person.
There is such a small amount of historic record for Joshua Badger I and II, and the fact that Joshua Badger I was an orphan and Joshua II was not a property owner and both of them dropped out of the public record while still a very young person has made this task extra difficult.
I welcome any help in this effort.
Thanks for the input.