I suspect you are seeing two things. First, you are once again inserting the ON resistance into the part of the circuit that produces the measurement, which desensitizes the measurement. You want the MUX on the high side of the thermistors. Second, you are probably reducing the current output of your current source because of a low output impedance. We have already determined that the resistance of the MUX channels is about 500Ω and that, near room temperature, the resistance of the thermistor is about 120Ω, or 620Ω total. If you are only seeing 760mA across the combination, then you only have about 1.1mA of current.

 

............

However, when I then connect the thermistor to one of my MUX inputs and the MUX output to earth (so I now have the MUX channel in series), the readings change to the following.

Cool - 780mV
Normal - 760mV
Warm - 730mV

............

Why are you connecting the MUX output to earth?

You connect one mux to switch the current source, as you initially showed, and another mux to switch the thermistor voltage to a high impedance voltage measurement circuit. That way the mux resistance is taken out of the measurement.

If you want a wider voltage range you can add an op amp non-inverting buffer amp with gain and an added negative offset voltage (positive voltage to the inverting input) to amplify the desired voltage change.

 

Why are you connecting the MUX output to earth?

Because that's how I connected the thermistor in my previous test...so I just 'inserted' the MUX to allow me to control which thermistor I was measuring.

You connect one mux to switch the current source, as you initially showed, and another mux to switch the thermistor voltage to a high impedance voltage measurement circuit. That way the mux resistance is taken out of the measurement.

Ha! The penny has finally dropped! I now understand what you have been suggesting. Apologies for being a bit slow
I've attached a sketch to confirm that I've now got it.

 

Good.

Now, when you adjust your current source, do it with as much of the actual circuit intact as possible. Replace one of your thermistors with a fixed 120Ω resistor (or however close you can get) that is either tight-tolerance or that you have done a halfway decent 4-wire measurement on so that you know what it's actual resistance is pretty closely. Then adjust your current source to deliver the desired current into that load.

Somewhat ironically, the high ON resistance of the MUX channels will actually now help improve your measurements because they increase the output impedance of your current source and make it so that your actual current will vary by a smaller amount as the thermistor resistance changes.

 

Cool!

Thanks to all who contributed and helped get me to a working solution!