Hi fred:
I offer the following as a suggestions. Each switch has a clock. The problem then reduces to synchronizing the clocks. I will discuss that after dealing with getting the switches to all turn on together.
From your description I assume that the collection of switches are stationary with respect to each other, so there are no simultaneity issues. I also assume there is only one master switch, and when this is activated, the process for having all switches turn on together starts. (If there are multiple master switches, then a much more complicated process is needed.) I also assume the time required to send a signal between the master switch and the other switches are significantly different.
Let M be the master switch. Let T be the maximum of the time requites for a signal to travel from M to each of the other switches. When the master switch is activated, the switch's internal logic calculates the sum of the current time and T (and perhaps also a small safety increment), and transmits this time value to all the switches. Then all the switches turn on when each of their respective internal clocks agree with the transmitted time.
The easiest way to synchronize all the identical clocks is when they area are all close together. They can each be connected with a wire, all of idenetical lengths, to a common signal source. All the closkes are set to zero, and when they receive a signal via the wire all the clocks start to run. The the individual swithces and clocks can be moved to their respoective destinations, but at a speed suficiently slow that relatavistic effects that effect their "ticking" speeds are negligible.
If I have misunderstood the constrains you intended for this problem, I apologize.
Regards,
Buzz