A number of National Cemeteries contain wartime Confederate burials; i.e., soldiers who died while in Federal captivity. These burial grounds didn't become National Cemeteries until after the war. Otherwise you're right on target.
Here's a post which describes the removal of Confederate dead from the Gettysburg battleground.
Bodies were sent elsewhere for much the same reasons as the Indian removals in the late 1830s -- they weren't wanted where they were.

http://history-sites.com/cgi-bin/bbs53x/mscwmb/webbbs_config.pl?read=13417