An analysis of the influence of common modifiers on the kinetics of the curing process of epoxy-anhydride vitrimers was carried out. As common modifiers to enhance the “vitrimeric” nature of the material, zinc acetylacetonate as a transesterification catalyst and glycerol as a modifier of hydroxyl group content were chosen. The curing process of all obtained compositions was studied by differential scanning calorimetry (DSC) followed by the application of the isoconversional approach. It was shown that additives significantly affect the curing process. The resulting cured polymers were shown to be chemically recyclable by dissolution in the mixture of ethylene glycol and N-methylpirrolidone in a volume ratio of nine to one. The introduction of both zinc acethylacetonate and glycerol to the neat formulation led to a decrease in the dissolution time by 85.7% (from 35 h for the neat epoxy-anhydride formulation to 5 h for the modified formulation). In order to show the opportunity of the secondary use of recyclates, the mixtures based on the basic composition containing 10 wt. % of secondary polymers were also studied. The introduction of a recycled material to neat composition led to the same curing behavior as glycerol-containing systems.