Deep eutectic solvents (DESs) have shown great promise as alternative materials for electrochemical applications. Two prototypical DESs comprising of choline chloride (ChCl) and glycerol and urea respectively, were studied using broadband dielectric spectroscopy (BDS), dynamic mechanical spectroscopy (DMS), and differential scanning calorimetry (DSC). In ChCl and glycerol (glyceline), slow sub-α relaxations linked to nanoscale heterogeneity between the two species are observed with BDS and DMS spectra. With increasing ChCl concentration, the sub-α dielectric relaxation, that is coupled to ion diffusion, becomes faster relative to the structural relaxation, while the slower mechanical relaxation remains unaffected. In contrast, ChCl and urea (reline) feature a coordinated hydrogen bonded network where the salt ions interact strongly with urea molecules. With increasing salt concentration, an increase in rate of structural dynamics and dc conductivity is observed up to the eutectic composition. However, it is remarkable that just 3mol% more ChCl above the eutectic composition results in slower dynamics by up to one order of magnitude. The differences between glyceline and reline can be attributed to the drastic difference between the nature of the resultant hydrogen bonded networks associated with glycerol and urea. These results will be discussed within the framework of recent models of dynamics and transport in ion conducting liquids.