How changing physical constants and violation of local position invariance may occur?

Light scalar fields very naturally appear in modern cosmological models, affecting such parameters of Standard Model as electromagnetic fine structure constant $\alpha$, dimensionless ratios of electron or quark mass to the QCD scale, $m_{e,q}/\Lambda_{QCD}$. Cosmological variations of these scalar fields should occur because of drastic changes of matter composition in Universe: the latest such event is rather recent (redshift $z\sim 0.5$), from matter to dark energy domination. In a two-brane model (we use as a pedagogical example) these modifications are due to changing distance to "the second brane", a massive companion of "our brane". Back from extra dimensions, massive bodies (stars or galaxies) can also affect physical constants. They have large scalar charge $Q_d$ proportional to number of particles which produces a Coulomb-like scalar field $\phi=Q_d/r$. This leads to a variation of the fundamental constants proportional to the gravitational potential, e.g. $\delta \alpha/ \alpha = k_\alpha \delta (GM/ r c^2)$. We compare different manifestations of this effect. The strongest limits $k_\alpha +0.17 k_e= (-3.5\pm 6) * 10^{-7}$ are obtained from the measurements of dependence of atomic frequencies on the distance from Sun (the distance varies due to the ellipticity of the Earth's orbit).