  We focus on the class of cosmological models with a time-evolving vacuum
energy density of the form $\rho_\Lambda=C_0+C_1 H+C_2 H^2$, where $H$ is the
Hubble rate. Higher powers of $H$ could be important for the early inflationary
epoch, but are irrelevant afterwards. We study these models at the background
level and at the perturbations level, both at the linear and at the nonlinear
regime. We find that those with $C_0=0$ are seriously hampered, as they are
unable to fit simultaneously the current observational data on Hubble expansion
and the linear growth rate of clustering. This is in contrast to the $C_0\neq
0$ models, including the concordance $\Lambda$CDM model. We also compute the
redshift distribution of clusters predicted by all these models, in which the
analysis of the nonlinear perturbations becomes crucial. The outcome is that
the models with $C_0=0$ predict a number of counts with respect to the
concordance model which is much larger, or much smaller, than the $\Lambda$CDM
and the dynamical models with $C_0\neq 0$. The particular case
$\rho_\Lambda\propto H$ (the pure lineal model), which in the past was
repeatedly motivated by several authors from QCD arguments applied to
cosmology, is also addressed and we assess in detail its phenomenological
status. We conclude that the most favored models are those with $C_0\neq 0$,
and we show how to discriminate them from the $\Lambda$CDM.

Cosmology and Nongalactic Astrophysics;General Relativity and Quantum Cosmology;High Energy Physics - Phenomenology