We explore the use of normalizing flows to emulate Monte Carlo detector simulations of photon showers in a high-granularity electromagnetic calorimeter prototype for the International Large Detector (ILD). Our proposed method - which we refer to as "Layer-to-Layer-Flows" (L2LFlows) - is an evolution of the CaloFlow architecture adapted to a higher-dimensional setting (30 layers of 10×10 voxels each). The main innovation of L2LFlows consists of introducing 30 separate normalizing flows, one for each layer of the calorimeter, where each flow is conditioned on the previous five layers in order to learn the layer-to-layer correlations. We compare our results to the BIB-AE, a state-of-the-art generative network trained on the same dataset and find a significantly improved fidelity of our model.