We consider a velocity field with linear viscous interactions defined on a one dimensional lattice. Brownian baths with different parameters can be coupled to the boundary sites and to the bulk sites, determining different kinds of non-equilibrium steady states or free-cooling dynamics. Analytical results for spatial and temporal correlations are provided by analytical diagonalisation of the system’s equations in the infinite size limit. We demonstrate that spatial correlations are scale-free and timescales become exceedingly long when the system is driven only at the boundaries. On the contrary, in the case a bath is coupled to the bulk sites too, an exponential correlation decay is found with a finite characteristic length. This is also true in the free cooling regime, but in this case the correlation length grows diffusively in time. We discuss the crucial role of non-homogeneous energy injection for long-range correlations and slow timescales , proposing an analogy between this simplified dynamical model and recent experiments with dense vibro-fluidized granular materials. Several generalizations and connections with the statistical physics of active matter are also suggested.