Systems consolidation refers to the reorganization of memory over time across brain regions. Despite recent advancements in unravelling engrams and circuits essential for this process, the exact mechanisms behind engram cell dynamics and the role of associated pathways remain poorly understood. Here, we propose a computational model to address this knowledge gap that consists of a multi-region spiking recurrent neural network subject to biologically-plausible synaptic plasticity mechanisms. By coordinating the timescales of synaptic plasticity throughout the network and incorporating a hippocampus-thalamus-cortex circuit, our model is able to couple engram reactivations across these brain regions and thereby reproduce key dynamics of cortical and hippocampal engram cells along with their interdependencies. Decoupling hippocampal-thalamic-cortical activity disrupts engram dynamics and systems consolidation. Our modeling work also yields several testable predictions: engram cells in mediodorsal thalamus are activated in response to partial cues in recent and remote recall and are crucial for systems consolidation; hippocampal and thalamic engram cells are essential for coupling engram reactivations between subcortical and cortical regions; inhibitory engram cells have region-specific dynamics with coupled reactivations; inhibitory input to mediodorsal thalamus is critical for systems consolidation; and thalamocortical synaptic coupling is predictive of cortical engram dynamics and the retrograde amnesia pattern induced by hippocampal damage. Overall, our results suggest that systems consolidation emerges from concerted interactions among engram cells in distributed brain regions enabled by coordinated synaptic plasticity timescales in multisynaptic subcortical-cortical circuits.