If they use AC synchronous motors, they will stay in sync. If they are quartz crystal DC powered clocks I imagine they share one power supply to avoid having to change batteries. Either way, DST changes require only unplugging the power supply for one hour or 23 hours. I used to have to sync dozens of clocks twice a year at 2AM in a TV station. Now we use timecode.

The National Electric Code states that for low voltage (less than 30 V) lighting systems the components must be UL listed and the bare conductors shall not be less than 2.1 meters (7 feet) above the floor. It's probably safe, but it isn't to code.