They can move a little (like a limbless person, I guess) according to the novel Queen of Blades. But in-game, they couldn't move at all.
They aren´t mobile themselves, but they can be transported by other means. Like a portable bridge or something.
I've always wondered, how the hell do overlords transport things? Do they shove every zerg into their intestines with their tentacles, and crap them out when they want to unload? Sounds pretty Disgusting
According to the research name in stacraft 1, they grow some sort of sacs on their belly And I think the transport of cerebrates would be left to the big zerg transports.
they used their behemoths to fly through a warp hole. I assume that if anyone is gonna transport a cerebrate completely, it'd take a behemoth.