Gone south had more than one meaning. Some southern sympathizers went south of their own accord, copperheads (read as, active confederate supporters in the north) were arrested & escorted through the union lines south.
We currently have our own modern copperheads (read as, leftie pols & LMSM terr perp supporters) they should be escorted out of the country, say, to Syria or Iran, where they would be more comfortable with their own kind.
Neocopperheads is the term for today's war underminers.
A Copperhead in New York had far more rights---indeed, far more chance of LIVING---than did a Union sympathizer in, say, Charleston. Even before the war, Unionists (especially if they mentioned slavery) were beaten, tarred and feathered, and jailed. Pro-confed. material could circulate freely in the North, but pro-Union material was banned by southern mails. While criticism of southern generals was tolerated in southern papers, there was NO criticism of the "cause" allowed, at all.
Heading south, Breckenridge joined the Confederate Army, rising to the rank of Major General (and he did quite well) before accepting a posting as Secretary of War just a few months before Richmond fell.
This is just an abbreviated version of the events, there are a lot more details, but the basic gist is the same.