The temporal evolution of entanglement between a noisy system and an ancillary system is analyzed in the context of continuous-time open quantum system dynamics. Focusing on a couple of analytically solvable models for qubit systems, we study how Markovian and non-Markovian characteristics influence the problem, discussing in particular their associated entanglement-breaking regimes. These performances are compared with those one could achieve when the environment of the system is forced to return to its input configuration via periodic instantaneous resetting procedures.