Systems consolidation is a common feature of learning and memory systems, in which a long-term memory initially stored in one brain region becomes persistently stored in another region. We studied the dynamics of systems consolidation in simple circuit architectures modeling core features of many memory systems: an early- and late-learning brain region and two sites of plasticity. We show that the synaptic dynamics of the circuit during consolidation of an analog memory can be understood as a temporal integration process, by which transient changes in activity driven by plasticity in the early-learning area are accumulated into persistent synaptic changes at the late-learning site. This simple principle leads to two constraints on the circuit operation for consolidation to be implemented successfully. First, the plasticity rule at the late-learning site must stably support a continuum of possible outputs for a given input. We show that this is readily achieved by heterosynaptic but not standard Hebbian rules, that it naturally leads to a speed-accuracy tradeoff in systems consolidation, and that it provides a concrete circuit instantiation for how systems consolidation solves the stability-plasticity dilemma. Second, to turn off the consolidation process and prevent erroneous changes at the late-learning site, neural activity in the early-learning area must be reset to its baseline activity. We propose two biologically plausible implementations for this reset that suggest novel roles for core elements of the cerebellar circuit.