Astrophysical tests of the stability of Nature's fundamental couplings are a key probe of the standard paradigms in fundamental physics and cosmology. In this report we discuss updated constraints on the stability of the fine-structure constant α and the proton-to-electron mass ratio μ=mp/me within the Galaxy. We revisit and improve upon the analysis by Truppe et al. [1] by allowing for the possibility of simultaneous variations of both couplings and also by combining them with the recent measurements by Levshakov et al. [2]. By considering representative unification scenarios we find no evidence for variations of α at the 0.4 ppm level, and of μ at the 0.6 ppm level; if one uses the [2] bound on μ as a prior, the α bound is improved to 0.1 ppm. We also highlight how these measurements can constrain (and discriminate among) several fundamental physics paradigms.