Definition 1 : to tour through rural districts staging usually theatrical performances 2 : to travel from place to place making brief stops (as in a political campaign or a promotional tour) 3 : to pilot one's airplane in sightseeing flights with passengers or in exhibition stunts in an unscheduled course especially in rural districts 4 : to travel across (an area) while barnstorming
Over the century and a half or so that barnstorm has been in use, the term has been employed to talk about theatrical performances that tour through rural districts, to promotional multi-stop trips by authors, politicians, and the like, and to performances in which pilots do stunts to show off their flying skills. Not all of these happen in rural areas, but they are often associated with the land of barns and fields. The storm part of barnstorm, though, is less idyllic. It's the same storm used to talk about troops storming a fort: an attack with a bold, swift frontal movement intended to defeat or win over quickly.