Within the context of covalent adaptable networks (CANs), we developed in this study a novel methodology to provide non-isocyanate polyurethane-based CANs with embedded tertiary amines that serve as internal catalytic moieties for the dynamic bond exchange processes. For the CAN design, we made use of multifunctional N-substituted 8-membered cyclic carbonates that are ring-opened by macromolecular amines. Several model reactions were conducted to investigate transcarbamoylation bond exchange reactions at elevated temperatures and assess the influence of catalytic moieties within the urethane structure. This led us to design a non-isocyanate polyurethane CAN wherein the position of the internal catalyst was changed with respect to previous reported polyhydroxyurethane CANs, while maintaining close proximity to the dynamic carbamate linkages. It is shown that this positioning change of the tertiary amines still resulted in an internal catalytic effect on the dynamic exchange reactions, hence providing a better understanding of the role of tertiary amines as internal catalysts during the reprocessing of polyurethane networks. Moreover, the model experiments and thermomechanical investigations of the (re)processed networks allowed to identify the occurrence of competing reactions, involving a dissociative mechanism giving rise to urea formation, which significantly impacts the materials’ reprocessability and properties.