The Yellow Vests have been coming out to demonstrate in Paris and other large French cities for 13 straight Saturdays now, bedeviling the government of President Emmanuel Macron with the vagueness of their demands and the lack of a leadership to negotiate with. How it plays out could have consequences well beyond France’s borders.

The size of the protests has been shrinking, and Mr. Macron’s approval ratings have been creeping back up from a devastatingly low 23 percent in December, after the demonstrators first emerged, initially to protest a rise in the tax on gasoline, which already costs more than $6 a gallon in France. But the Yellow Vests show no sign of ending their weekly invasions of the capital anytime soon, and polls show that a majority of the French continue to support them.

Their protests have morphed into a popular movement, an uprising of provincial towns and villages — what urban French idealize as “la France profonde,” the deep, timeless France — against a sense of being forgotten in their picturesque countryside with incomes that barely stretch to the end of each month.

Though organized protests by unions, students or other groups of the left or right are a fixture of French public life, the Yellow Vests are something new and unfamiliar in their absence of an organization, defined demands or ideology. Violent fringe groups have latched onto the weekly protests, clashing with the police, setting cars on fire and smashing store windows, but a large majority of the Yellow Vests are neither violent nor radical.