  Constraints on the cosmological concordance model parameters from observables
at different redshifts are usually obtained using the locally measured value of
the gravitational constant $G_N$. Here we relax this assumption, by considering
$G$ as a free parameter, either constant over the redshift range or dynamical
but limited to differ from fiducial value only above a certain redshift. Using
CMB data and distance measurements from galaxy clustering BAO feature, we
constrain the cosmological parameters, along with $G$, through a MCMC bayesian
inference method. Furthermore, we investigate whether the tensions on the
matter fluctuation $\sigma_8$ and Hubble $H_0$ parameter could be alleviated by
this new variable. We used different parameterisations spanning from a constant
$G$ to a dynamical $G$. In all the cases investigated in this work we found no
mechanism that alleviates the tensions when both CMB and BAO data are used with
$\xi_{\mathrm{g}} = G / G_N$ constrained to 1.0$\pm0.04$ (resp. $\pm0.01$) in
the constant (resp. dynamical) case. Finally, we studied the cosmological
consequences of allowing a running of the spectral index, since the later is
sensitive to a change in $G$. For the two parameterisations adopted, we found
no significant changes to the previous conclusions.
