Have there been studies of "old photons" to see just how constant things like Planck constant has been? The question Are photons aged? and answers therein have got me thinking:

I vaguely remember hearing something about experiments where "old photons" were collected by large telescopes from very distant objects, and either their energies or their frequencies were compared to their wavelengths to check the (linear or inverse) some fundamental constants, perhaps those used in:

$$E = hf = h \frac{c}{\lambda}$$

Questions:


Am I misremembering, or has this been done?
What were the results and to what level of precision was non-variation constrained?
You can not check if a dimensional constant has changed because you can always reverse that change by a smart change of coordinates (system of units). Despite that, since the current Physics assumes the immutability of certain constants, you can verify this assumption by testing the change of an adimensional constant.

One of the most common adimensional constants that is used to search for constant variability if the fine-structure constant (a fundamental physical constant characterizing the strength of the electromagnetic interaction between elementary charged particles)
$$\alpha=\dfrac{1}{4\pi\epsilon_0}\dfrac{e^2}{\hbar c}\,.$$

The Wikipedia pages about Time-variation of fundamental constants and fine-structure constant have nice explanations about this.