That would probably involve relays (ugh!) most analog muxes are concerned only with getting voltages through with low offsets and errors. You could terminate the loops at the mux and then mux the resulting voltages and use a voltage to current loop translator at the output. This seems like it would be quite reasonable to do if it would meet your accuracy requirements. You would want differential inputs to prevent grounding issues and would need to watch common mode voltages, but it should work.
You would almost have to use this approach anyway. While it is quite possible to mux currents with an appropriate discrete design, most loop entities indicate faults when you open the loop and switching off channels to a resistor to maintain loop current would generate noise and slew rate/settling time issues.
Often if a gizmo doesn't exist, it's not because it's not possible but that it's not a good idea and would be hard to apply. The terminate/mux/retransmit idea would be my recommendation. It might also be cost competitive to simply set up a showbox PC with a 16 channel analog card
and a D/A output and do the deed digitally. Or better, just send the results on the network which would solve the problem of having to demux at the other end. All factors considered I'll bet the PC would be cheaper.
Unless I'm not fully understanding the issue, an analog mux chip would work great. I pulled open Digikey and found a 16 channel DIP from analog devices that muxes 16 channels. I'm sure if you follow their reference schematic, you'll be up in running in no time. Digikey part#: AD7506JN-ND This is also the AD part number if you take off the "-ND" suffix.
If you can tolerate a certain addition of impedance, an analog mux chip would work fine. Usually these chips add 50-150 ohms of series resistance to your signal in the "on" state. So, if you are driving a high impedance device like a data aquisition card, it works great. In fact almost all ADC cards use these same devices to get multiple channels.
If, however, you want to drive a low impedance, that is another story, and something like a relay or a more sophisticated (and complicated) analog chip would be needed.
Is this design a 4-20ma kind of system, or are you muxing voltage? I was a bit confused by the last email.