In mammals, repression of translation during stress is associated with the assembly of stress granules in the cytoplasm, which contain a fraction of arrested mRNA and have been proposed to play a role in their storage. Because physical contacts are seen with GW bodies, which contain the mRNA degradation machinery, stress granules could also target arrested mRNA to degradation. Here we show that contacts between stress granules and GW bodies appear during stress-granule assembly and not after a movement of the two preassembled structures. Despite this close proximity, the GW body proteins, which in some conditions relocalize in stress granules, come from cytosol rather than from adjacent GW bodies. It was previously reported that several proteins actively traffic in and out of stress granules. Here we investigated the behavior of mRNAs. Their residence time in stress granules is brief, on the order of a minute, although stress granules persist over a few hours after stress relief. This short transit reflects rapid return to cytosol, rather than transfer to GW bodies for degradation. Accordingly, most arrested mRNAs are located outside stress granules. Overall, these kinetic data do not support a direct role of stress granules neither as storage site nor as intermediate location before degradation.