Cases of any known fundamental physical constants changing within our locality?
There have been some attempts to "locally" (i.e. non-astronomically) measure the drift of the fine-structure constant $\alpha=e^2/\hbar c$, using experiments on trapped atoms and ions. If you want to read up on it I think this review article,
Lea, S. N. Limits to time variation of fundamental constants from comparisons of atomic frequency standards. Rep. Prog. Phys. 70 no. 9, p. 1473 (2007). doi:10.1088/0034-4885/70/9/R01.
is a good place to start.
I know of one example (of which I can't find a good up-to-date, readable reference, possibly because the experiment may still be ongoing), currently taking place at NPL. This uses an octupole transition in ytterbium: an S$\rightarrow$F transition which is forbidden by both dipole and quadrupole selection rules. This makes the excited state have a very long lifetime - six years - which makes the resonance very narrow in comparison to the ~470 nm wavelength. This makes the transition very valuable for precision spectroscopy.
[Edit: here's a good reference: NPL | Physics | PRL | arXiv.]
Even better, this transition can be compared to a quadrupole transition in ytterbium ions, which is not as narrow but is still good enough for metrology purposes. The real catch is that the details of the frequencies of these two E2 and E3 transitions depend on $\alpha$ in opposite directions, so any change in it will make the frequency ratio change.
At the current state of the art in these experiments, the precision achievable is enough that any "astronomical-scale" changes in $\alpha$ (i.e. of the order of 1% per billion years) should be observable by measuring these kind of frequency ratios over a few years. This is an active area of research and experiments are currently running which will report in the near (i.e. ~few years) future, with precision matching or exceeding current astronomical bounds on $d\alpha/dt$.
I guess from your use of the phrase "within our locality" that this is not what you mean, but various people have tried to measure astrophysically whether fundamental constants have evolved over time. A while back, there were some claims (e.g., this paper and others by the same people) that observations of spectral lines in very distant objects could best be explained by assuming that fundamental constants (specifically the fine-structure constant) had changed with time. There is, to say the least, no consensus that these observations and their interpretation are correct, though.
People certainly try to measure changes in physical constants locally, but as far as I know there's no evidence for such variation.
It's somewhat glib to put quantum mechanics this way, but one could say that Planck's constant was zero before Planck introduced it, insofar as the classical limit of QM is $\hbar\rightarrow 0$ (which is not so very far).
Something I've played with, to absolutely no good effect so far, is the idea that quantum fluctuations might be greater or less from place to place and from time to time, which fits a little better with your Question as you have put it. However, one would then describe quantum fluctuations at different space-time points relative to the unchanged constant $\hbar$, which would now be defined as "the amplitude of quantum fluctuations under such and such conditions". That's essentially Solomoan's Answer, however his apparent assumption that the idea is obvious is a little too fast for someone with almost any philosophical leanings. [The boiling point of water is a constant, 100°C, if one defines the conditions carefully enough. A very interesting account of the historical and philosophical process associated with the definition of temperature scales is given by Hasok Chang, in a book that won a major Philosophy of Science prize, "Inventing Temperature". If you read that —it's very accessible as these things go— you will ask a different Question.]
One way to approach this Question may be to suppose that new theories introduce new constants, in terms of which old constants can be expressed, together with the conditions in which they accurately describe the Physical world. Thus, in my speculative example as I put it above, one might define the amplitude of quantum fluctuations as $\hbar$ at the surface of the earth, and describe it's variation as a function of altitude —this would at least change our theory of gravity, and likely lots else besides, so the whole of Physics becomes part of the discussion. That is a thread in Philosophy of Science that is known as the Quine-Duhem thesis, or, as Wikipedia has it, the Duhem-Quine thesis, which can be loosely summarized as the idea that a theory stands or falls as a whole, a point that I take to be vitiated by the always present possibility of changing any theory in very small ad-hoc ways.
Your Question to some extent opens a philosophical Pandora's box, which you should be cautious about opening. My view is that anyone who wants to change Physics significantly must consider these and similar ideas at length, but for Physicists trying to do the everyday job of Physics it is perhaps as counterproductive to spend time on this as it is to spend time learning to play the violin.