The non-equilibrium dynamics of linear and star-shaped cis-1,4 polyisoprenes confined within nanoporous alumina is explored as a function of pore size, d, molar mass, and functionality (f = 2, 6, and 64). Two thermal protocols are tested: one resembling a quasi-static process (I) and another involving fast cooling followed by annealing (II). Although both protocols give identical equilibrium times, it is through protocol I that it is easier to extract the equilibrium times, teq, by the linear relationships of the characteristic peak frequencies with time and rate, respectively, as log(fmax) = C1 - k log(t) and log(fmax) = C2 + λ log(β). Both thermal protocols establish the existence of a critical temperature (at Tc, where k → 0 and λ → 0) below which non-equilibrium effects set-in. The critical temperature depends on the degree of confinement, 2Rg/d, and on molecular architecture. Strikingly, establishing equilibrium dynamics at all temperatures above the bulk, Tg, requires 2Rg/d ∼ 0.02, i.e., pore diameters that are much larger than the chain dimensions. This reflects non-equilibrium configurations of the adsorbed layer that extent away from the pore walls. The equilibrium times depend strongly on temperature, pore size, and functionality. In general, star-shaped polymers require longer times to reach equilibrium because of the higher tendency for adsorption. Both thermal protocols produced an increasing dielectric strength for the segmental and chain modes. The increase was beyond any densification, suggesting enhanced orientation correlations of subchain dipoles.