  We compute the time variation of the fundamental constants (such as the ratio
of the proton mass to the electron mass, the strong coupling constant, the fine
structure constant and Newton's constant) within the context of the so-called
running vacuum models (RVM's) of the cosmic evolution. Recently, compelling
evidence has been provided showing that these models are able to fit the main
cosmological data (SNIa+BAO+H(z)+LSS+BBN+CMB) significantly better than the
concordance $\Lambda$CDM model. Specifically, the vacuum parameters of the RVM
(i.e. those responsible for the dynamics of the vacuum energy) prove to be
nonzero at a confidence level $\gtrsim3\sigma$. Here we use such remarkable
status of the RVM's to make definite predictions on the cosmic time variation
of the fundamental constants. It turns out that the predicted variations are
close to the present observational limits. Furthermore, we find that the time
variation of the dark matter particles should be crucially involved in the
total mass variation of our Universe. A positive measurement of this kind of
effects could be interpreted as strong support to the "micro and macro
connection" (viz. the dynamical feedback between the evolution of the
cosmological parameters and the time variation of the fundamental constants of
the microscopic world), previously proposed by two of us (HF and JS).
