how does an ADC cope with a varying analogue voltage?

Yes, kind of but not in the way you may think.

Both conversion rates are very slow compared to the sampling type of conversions. The conversion actually integrates (averages) the signal over the conversion time.

So for the 10 Hz rate (we will forget about the communication time in between conversions here) the input is averaged over 100 ms. Because the averaging is essentially analog it excels in noise cancellation. This is why they can make a claim for a 24 bit conversion. see (1)

So if you are planning on taking multiple samples and averaging them you would be better with the slower rate as it has 8X the time to average compared to the higher rate.

Let me know if this makes sense to you.

(1) 24 bits at a gain of 128 means each bit is 0.6µV not an easy task.