Dynamic-covalent chemistry has enabled the facile synthesis of a new generation of degradable materials, but controlling the rate at which these materials degrade remains elusive. Using segmented hyperbranched polymers (SHPs) as model branched architectures, we demonstrate that SHPs containing imine cross links degrade under acidic conditions into well-defined linear chains at rates controllable via modification of the imine N-substituent. Imine-cross-linked SHPs were synthesized in a one-pot protocol by reversible addition fragmentation chain transfer (RAFT) copolymerization of novel divinyl compounds containing dynamic covalent oxime, semicarbazone, and acyl hydrazone moieties. The extent of SHP branching could be controlled through the relative stoichiometric ratios of cross linker and chain transfer agent (CTA), and studies of the polymerization kinetics confirmed the growth of polydisperse branched species at high monomer conversions. When subjected to aqueous acidic conditions, the polydisperse branched architecture degraded into well-defined polymers, a process that was accelerated under more strongly acidic conditions and by incorporating less hydrolytically stable imine cross-links. Finally, we found that the rate of SHP degradation could be tuned with an unprecedented level of control by cross-linking the polymers with different proportions of multiple imines.