  Running Dark Energy and Dark Matter models are candidates to resolve the
Hubble constant tension. However the model does not consider a Lagrangian
formulation directly. In this paper we formulate an action principle where the
Running Vacuum Model (RVM) is obtained from an action principle, with a scalar
field model for the whole dark components. The Dynamical Spacetime vector field
$\chi_\mu$ is a Lagrange multiplier that forces the kinetic term of the scalar
field to behave as the modified dark matter. When we replace the vector field
by a derivative of a scalar the model predicts diffusion interactions between
the dark components with a different correspondence to the RVM. We test the
models with the Cosmic Chronometers, Type Ia Supernova, Quasars, Gamma ray
Bursts and the Baryon Acoustic Oscillations data sets. We find that
$\Lambda$CDM is still the best model. However this formulation suggests an
action principle for the $\Lambda$CDM, the RVM model and other extensions.

General Relativity and Quantum Cosmology