This was a young adult novel about a group of children who are compelled to walk from New England to D. C. as a form of protest (maybe echoes of nuclear war?). One of the kids is, I think, the granddaughter of the president but keeps that a secret. As the march continues, more children join in. There’s some sort of mystical element. I read the book in the mid-1980s.
Sounds like The Fragile Flag by Jane Langton. It’s part of a series that includes The Diamond in the Window and The Fledgling (which won the Newbery Medal)
Is it The Fragile Flag by Jane Langton?
https://books.google.com/books?id=gfNjEbJZTtMC&printsec=frontcover&dq=children+march+washington+nuclear+book+protest&hl=en&sa=X&ved=0ahUKEwiCtuahh_7SAhXMQiYKHbXsCV0Q6AEIHDAA#v=onepage&q&f=false
This sounds a lot like The Long Walk by Stephen King. Although the main character is secretly the son of the General (in charge of The Walk). Many of the elements you describe are the same.
I’m virtually sure it’s The Fragile Flag (1984).
Hi, I’m the original poster. It absolutely is The Fragile Flag. Thank you all so much!