Deep eutectic solvents have recently gained significant attention as versatile and inexpensive materials with many desirable properties and a wide range of applications. In particular, their characteristics, similar to those of ionic liquids, make them a promising class of liquid electrolytes for electrochemical applications. In this study, we utilized a local equivariant neural network interatomic potential model to study a series of deep eutectic electrolytes based on lithium bis(trifluoromethanesulfonyl)imide (LiTFSI) using molecular dynamics (MD) simulations. The use of equivariant features combined with strict locality results in highly accurate, data-efficient, and scalable interatomic potentials, enabling large-scale MD simulations of these liquids with first-principles accuracy. Comparing the structure of the liquids to the reported results from classical force field (FF) simulations indicates that ion-ion interactions are not accurately characterized by FFs. Furthermore, close contacts between lithium ions, bridged by oxygen atoms of two amide molecules, are observed. The computed cationic transport numbers (t+) and the estimated ratios of Li+-amide lifetime (τLi-amide) to the amide's rotational relaxation time (τR), combined with the ionic conductivity trend, suggest a more structural Li+ transport mechanism in the LiTFSI:urea mixture through the exchange of amide molecules. However, a vehicular mechanism could have a larger contribution to Li+ ion transport in the LiTFSI:N-methylacetamide electrolyte. Moreover, comparable diffusivities of Li+ cation and TFSI- anion and a τLi-amide/τR close to unity indicate that vehicular and solvent-exchange mechanisms have rather equal contributions to Li+ ion transport in the LiTFSI:acetamide system.