I realize this thread is about a year old, but having recently designed a robust monophonic instrument tuner, I feel somewhat qualified to comment on this.
First, let’s consider a time domain approach using cross-correlation. This is common as far as I can tell, at least in the published literature. Let’s make the following assumptions for the sake of illustration: an Fs of 48000, and a desired tuning range of 30.9Hz (B0) to say 880Hz (A5). Rounding, there are then 1559 samples/period at B0, and 54.5 samples/period at A5. This appears to show higher resolution for low frequencies as there are more samples per period. In practice you would calculate the cross-correlation with enough lags to include at least 1559 samples, and probably a bit more to have some margin. Within that time period, you would have 28.6 periods of A5, so you could identify 28 peaks that would occur, and divide the last peak location (in samples) by 28 to get a time resolution almost as good as the lower frequency. So, I would say that using the cross-correlation approach, the precision is roughly constant, in the sense of the question posed.
1 cent is ~578ppm. Translating the above into cents gives .90 samples/cent at B0 and .032 samples/cent at A5. Interpolation is clearly needed, even if multiple peaks are used to increase the resolution for the higher frequencies. I think the two main problems with the cross-correlation approach are 1) the cost of calculation the lags and 2) getting accurate results on the lowest frequencies when the tones are very pure. 1) can be addressed using and FFT/IFFT approach as an alternative to direct form convolution to calculate the lags. 2) is a stickier problem. Tones with more complex overtones have better defined peaks because the fingerprint that slides through the correlation is more complex, resulting in higher output when matching and lower output when mismatching.
Let’s now consider a frequency domain approach using an FFT. This is not common as far as I can tell. All the sources that I’ve found say that in practice you can’t get enough resolution to make this work. Using a FFT of size 16384, there are 10.5 frequency bins/cycle at B0 and 300 bins/cycle at A5. Going to cents, we get .0061 bins/cent at B0 and .174 bins/cent at A5. There is clearly more resolution at the higher frequencies, and is in keeping with the OP’s understanding as the question was posed.
Trying to use quadratic interpolation with these low levels of initial precision does not work well at all. However, I have found a way to easily get around this problem. Let’s consider that the highest frequency we are trying to measure is indeed the A5 at 880Hz. With an Fs of 48000, the Nyquist frequency is 24KHz, but we don’t need all of that. We can use a single biquad IIR lowpass set to 900Hz to filter a time slice of 8096 samples. Then we can decimate by a factor 25, leaving just 324 time samples (every 25th one). We can then zero pad those to a length 16384 and take an FFT. This results in another resolution boost of 2 (8K->16K) for a total resolution boost factor of 50. With these methods, our resolution becomes 0.3 bins/cent at B0 and 8.7 bins/cent at A5. This is way more than we need in the higher range, and just a bit shy in the lower range, so we can use a quadratic interpolator and make short work of it. Another benefit of this approach is that pure tones work perfectly. More complex tones do as well, but in the real-world cases where the harmonics can be louder than the fundamental, it’s not as simple as just looking for the tallest bin magnitude. A bit more shenanigans is required to discern which spectral peak represents the fundamental tone. It turns out to be much easier to do this in the frequency domain vs. in the time domain trying to sort out the peaks in the lags.
So, to sum up the answer to the OP's question - it might be dynamic or it might not be dynamic, depending on the approach taken. And if it is, it might not be visible to a user if the resolutions across frequencies are all below the minimum that would matter.
