It's an interesting question, but I think it's a poorly worded one since there is a strong argument to be made that Dylan was never involved in politics in the first place. Dylan got his start in the folk scene of New York, playing the various coffee shops and clubs of the Greenwich Village part of New York. Folk, especially the type sung by one of Dylan's major influences Woody Guthrie, had a populist element to it to that Dylan emulated in his own early works, and you can see this especially echoed in Dylan's Talkin... Blues songs which were directly influenced by Talkin' Dust Bowl Blues.

However, Dylan's political commentary was rather strictly contained to his music and songwriting. It was the movement that made Dylan the figurehead, rather than Dylan rising up to lead it. To wit, Dylan's songs of the time as as much commentaries on the world at large as they are on the movement that claimed him as their leader. For Dylan the content was more a result of the genre than ideology, and you can see he had a lot of interest in not just Folk but also early Blues. His first album contains more Blues standards than it does Folk or Folk influenced ones.

By the mid-60's it seemed Dylan had moved on musically. Ditching "politics" wasn't so much that as it was simply moving on to something else he found interesting. The folk movement rather infamously chastised him for this, known as the period when Dylan "went electric," at the 1964 1965 Newport Folk Festival.

Moreover, I would also argue that the bulk of Dylan's songwriting of the "political" period isn't actually political. It's commentary, for certain, but part of the genius of Dylan is that it's a timeless commentary, lyrics that speak to something more primal about society.

Please forgive me for what I think might be rambling, I'm going from memory on a lot of this. Dylan is a complex subject, and people have dedicated a lot of time to studying both it and him. I think you'll find, though, that as a person his interests lie with music, and he lets the music speak for its self. Dylan didn't attend protests, he didn't claim to speak for a movement, and he was notorious for being coy and cold in interviews he gave to the press.

While there are a number of books on the subject, I would recommend starting with the documentary No Direction Home, which covers this period in Dylan history quite extensively and includes examples of these past interviews. Joan Baez, if I recall, even mentions this idea that Dylan was uninterested in the movement specifically, as she expressed a bit of frustration that Dylan didn't want much to do with it. (As a side note, this is also the root of the strife that lead to the "going electric" reaction. The folk movement had adopted Dylan, and playing electric was seen as a direct rejection of them. Not to mention the song he was playing was Maggie's Farm, a song that is rather blatant about rejection).

Dylan wrote a song in 1965 called "My Back Pages" here he analyzed his reasons for turning away from his earlier political view of the world. See the lyrics below. I would also add that Dylan had two very political girlfriends in the early 1960s. Suze Rotolo and Joan Baez. After he broke up with each them his political involvement declined.

Crimson flames tied through my ears Rollin' high and mighty traps Pounced with fire on flaming roads Using ideas as my maps "We'll meet on edges, soon," said I Proud 'neath heated brow Ah, but I was so much older then I'm younger than that now.

Half-cracked prejudice leaped forth "Rip down all hate," I screamed Lies that life is black and white Spoke from my skull, I dreamed Romantic facts of musketeers Foundationed deep, somehow Ah, but I was so much older then I'm younger than that now.

Girls' faces formed the forward path From phony jealousy To memorizing politics Of ancient history Flung down by corpse evangelists Unthought of, thought, somehow Ah, but I was so much older then I'm younger than that now.

A self-ordained professor's tongue Too serious to fool Spouted out that liberty Is just equality in school "Equality," I spoke the word As if a wedding vow Ah, but I was so much older then I'm younger than that now.

In a soldier's stance, I aimed my hand At the mongrel dogs who teach Fearing not that I'd become my enemy In the instant that I preach My existence led by confusion boats Mutiny from stern to bow Ah, but I was so much older then I'm younger than that now.

Yes, my guard stood hard when abstract threats Too noble to neglect Deceived me into thinking I had something to protect Good and bad, I define these terms Quite clear, no doubt, somehow Ah, but I was so much older then I'm younger than that now.

For further information about this period of Dylan's life read this book: Rotolo, Suze (2009), A Freewheelin' Time, Aurum Press, ISBN 0-7679-2688-9