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Modern deep neural networks are powerful and widely applicable models that extract task-relevant information through multi-level abstraction. Their cross-domain success, however, is often achieved at the expense of computational cost, high memory bandwidth, and long inference latency, which prevents their deployment in resource-constrained and time-sensitive scenarios, such as edge-side inference and self-driving cars. While recently developed methods for creating efficient deep neural networks are making their real-world deployment more feasible by reducing model size, they do not fully exploit input properties on a per-instance basis to maximize computational efficiency and task accuracy. In particular, most existing methods typically use a one-size-fits-all approach that identically processes all inputs. Motivated by the fact that different images require different feature embeddings to be accurately classified, we propose a fully dynamic paradigm that imparts deep convolutional neural networks with hierarchical inference dynamics at the level of layers and individual convolutional filters/channels. Two compact networks, called Layer-Net (L-Net) and Channel-Net (C-Net), predict on a per-instance basis which layers or filters/channels are redundant and therefore should be skipped. L-Net and C-Net also learn how to scale retained computation outputs to maximize task accuracy. By integrating L-Net and C-Net into a joint design framework, called LC-Net, we consistently outperform state-of-the-art dynamic frameworks with respect to both efficiency and classification accuracy. On the CIFAR-10 dataset, LC-Net results in up to 11.9$\times$ fewer floating-point operations (FLOPs) and up to 3.3% higher accuracy compared to other dynamic inference methods. On the ImageNet dataset, LC-Net achieves up to 1.4$\times$ fewer FLOPs and up to 4.6% higher Top-1 accuracy than the other methods.