A new class of Friedmann-Lemaître-Robertson-Walker (FLRW) cosmological models with time-evolving fundamental parameters should emerge naturally from a description of the expansion of the universe based on the first principles of quantum field theory and string theory. Within this general paradigm, one expects that both the gravitational Newton’s coupling G and the cosmological term Λ should not be strictly constant but appear rather as smooth functions of the Hubble rate H(t). This scenario (“running FLRW model”) predicts, in a natural way, the existence of dynamical dark energy without invoking the participation of extraneous scalar fields. In this paper, we perform a detailed study of some of these models in the light of the latest cosmological data, which serves to illustrate the phenomenological viability of the new dark energy paradigm as a serious alternative to the traditional scalar field approaches. By performing a joint likelihood analysis of the recent supernovae type Ia data (SNIa), the Cosmic Microwave Background (CMB) shift parameter, and the Baryonic Acoustic Oscillations (BAOs) traced by the Sloan Digital Sky Survey (SDSS), we put tight constraints on the main cosmological parameters. Furthermore, we derive the theoretically predicted dark-matter halo mass function and the corresponding redshift distribution of cluster-size halos for the “running” models studied. Despite the fact that these models closely reproduce the standard ΛCDM Hubble expansion, their normalization of the perturbation’s power-spectrum varies, imposing, in many cases, a significantly different cluster-size halo redshift distribution. This fact indicates that it should be relatively easy to distinguish between the “running” models and the ΛCDM using realistic future X-ray and Sunyaev-Zeldovich cluster surveys.