I do not have any information on the Doe family, but my French Canadian family that lived in Pinnebog lived in Goedrich, Huron, Canada....before they crossed Lake Huron.
My family was sailors and probably had access to a boat, family lore. My family were into logging and there was a large forest fire, and that the family migrated to Michigan, Huron County for land and work. They evenutally migrated up to Alcona/Alpena county. I doubt that they were ever naturalized. My migrating relatives said that they were born in Louisana, Colorado, Canada, Michigan. They started with the correct information and it changed to Louisana, Colorado, and finally Michigan later in life to avoid the naturalization question...I think.
I have been told if you can get a look at this book that it would be helpful for my Canadian French research. It is on the canadian french family from Huron, Canada.
Also, I have been told that all of the church records for Huron, Canada is at the FHL Libraries. I have not made the trip yet to check it out.
I was told that all of my people came from Quebec but they migrated west looking for cheaper land as time went on. I saw a family tree done by someone else and between 1600's to 1850 they moved closer and closer to Michigan down the St. Lawerence waterway. They moved north in Michigan into the UP and west to Wisconsin and Minnesota. Eventually I found people in Alberta Canada because they had land grants for anyone that could settle the land and establish a farm.
Over time, they may have shortened and made their name more easily spoken for those non-french speaking people. My family went from DesRosiers to DeRosia.
You will want to do some reading on the Canadian French. I found lots of helpful post on ancestry.com.
I hope my scenario helps you and give you some insight.
Enjoy!
Yvonne