To the OP. As I said, you will need four VCAs (voltage controlled amplifiers). Google up some schematics. I would recommend using operational transconductance amplifiers for these, as it's fairly simple. They would all be driven from the same signal source, and each one would drive a different channel of your quadrophonic amplifier.
Generating the control signal is a bit more difficult I think.
I guess you only want to pan the sound between pairs of speakers - for example, start at front left and pan smoothly to front right, then pan smoothly to rear right, then pan smoothly to rear left, then pan smoothly to front left again and repeat.
In that case, no more than two channels are ever active at the same time, and you could use that fact to simplify the design. If you want the sound to move in a circular pattern, then three channels need to be active simultaneously, and that would complicate the design significantly I think.
With the two channel limit, you just need a circuit that will generate two output voltages (to drive two VCAs), that represent left and right volume values, that change over time to produce the effect of the signal moving smoothly from left to right and back. You then need some logic to switch each of those control voltages between two different VCAs at the appropriate times.
Creating the effect of a signal source moving smoothly from one speaker to another without changing its apparent volume is not as simple as it sounds, because of how the ear percieves volume, especially when the signal is coming from two separate sources. There are several approaches to panning, and I'm not well-versed on the subject. I suggest you Google some keywords like smooth pan audio left to right constant perceived volume. You may be able to find circuits that will do this.
You also need to make sure that the VCA's response to the control signals follows the correct "law". I believe an OTA-based VCA's voltage gain is directly proportional to the control current. So you need to find a pan circuit that generates control voltages (voltages can be converted into currents using op-amps) that are designed to drive an amplifier with linearly controlled voltage gain.
If you find some VCA and pan control circuit designs you like, post the links here and I'll try to suggest how you can connect them together.