No, they don't move. They are screwed to the back of the seat brace and the trunk floor. They were just supposed to function like a small speaker enclosure in the cavity of the trunk, since the sound chamber of the package shelf, glass, and roof of a coupe is not present to give dimension to the sound.

The original design was to cut through the bottom layer of vinyl that appears when the top is raised, and the speaker grills installed through that. It works okay if the top is up, but that's not how I drive my convertible. It's also not good for the speakers if you put your top down wet.

Quality in the '60s meant you had FM...not today's standards. The cars don't have much sound deadening to start with, the big engines contribute more to the "ambience" and the shape of the cars pushing through the air contributed to environmental noise. Trying to get new car audio quality without redesigning the car is futile. So enjoy what you can get for a reasonable expenditure, or go rustic and keep the original audio.

Mine still has the original AM/FM stereo installed, and a hidden speaker under the dash gives the "classic" sound when I feel in the mood, like in the judging lane at a local show.