The segmental dynamics of a low glass transition temperature T-g polymer under different conditions of morphological confinement are shown to manifest the influence of the frustration of the local "packing" of chain segments at interfaces of varying curvatures, unique to the morphology. Of particular interest are the polyisoprene (PI) segmental dynamics in polystyrene-b-polyisoprene (PS-b-PI) micelles and in layered onion-like structures in high T-g, "frozen," polystyrene (PS) hosts. At temperatures close to the T-g of PI domains, the segmental relaxation times of PI chains in the micellar structures, tau(micelle), are more than 1 order of magnitude shorter than those of PI chains in the onion-likestructures, tau(onion). These rates are appreciably faster than those of homopolymer PI and of PI in the neat copolymers possessing lamellar and gyroid phases, tau(homo) similar to 0.5 tau(lam) similar to 0.5 tau(gyroid), respectively. At high temperatures tau(homo) similar to tau(lam) similar to tau(gyroid) similar to tau(onion) similar to tau(micelle) These observations are readily reconciled with trends in the local T-g of the PI phase in each sample.