Digital or analog transmission for transmitting data off-board?
3.2 kHz is relatively low.  You are basically wanting to send audio signals around.  This can be done just like it would be with audio, which is usually with a shielded coax cable.  You don't say what accuracy you need, but a bigger problem may be offset caused by the power current in the ground line.  There are ways to overcome that too, but things get complicated with analog as you want higher accuracy or signal to noise ratio.

Another possibility to consider is to put a small microcontroller on the remote boards right with the sensor.  This can do whatever it has to to talk to the sensor.  It then sends the data back digitally.  Another advantage is that this micro can sample the sensor much more often than the minimum 6.4 kHz Nyquist rate, then do dessimation and send on the data close to the Nyquist rate.  This allows some range for your hardware anti-aliasing filter.

Let's say you send a new sample at 10 kHz rate and it takes 2 bytes.  Using a ordinary UART this would require a 20 bit times at 10 kHz rate, for a minimum baud rate of 200 kHz.  You aren't stuck with a standard baud rate when you own both ends.  Run it at 500 kBaud and you have plenty of overhead.  500 kBaud is quite possible over the "short distance" you specify.  Now you only need 3 wires in the cable, power, ground, and digital signal.