60 year old duct system with 4 inch ducts

My house used to have a low pressure GE oil furance. Nice alumn. ductwork, but all the runs to the boots are 4 inches. In 2001 the former owner put in a new Lennox oil furance. The fun cycled constantly, even though she had a smaller fan put in (5 inch). Now that I'm remodeling I want to switch to gas and was hoping to use the existing duct work. I understand that there MAY BE variable speed gas furnaces that will do OK with the 4 inch ducting? Some contractors have told me I don't have any options and that I must size up the ducting to 6 inch. I can't get to the majority of the ducting and I have lath and plaster, so I don't want to start tearing into the walls at this point. I don't want to purchase a new furnance and find out I am still going to have the same problem with the fan cycling etc. Any thoughts on this?