This work introduces a new training and compression pipeline to build nested sparse convolutional neural networks (ConvNets), a class of dynamic ConvNets suited for inference tasks deployed on resource-constrained devices at the edge of the Internet of Things. A nested sparse ConvNet consists of a single ConvNet architecture, containing <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> sparse subnetworks with nested weights subsets, like a Matryoshka doll, and can trade accuracy for latency at runtime, using the model sparsity as a dynamic knob. To attain high accuracy at training time, we propose a gradient masking technique that optimally routes the learning signals across the nested weight subsets. To minimize the storage footprint and efficiently process the obtained models at inference time, we introduce a new sparse matrix compression format with dedicated compute kernels that fruitfully exploit the characteristic of the nested weights subsets. Tested on image classification and object detection tasks on an off-the-shelf ARM-M7 microcontroller unit (MCU), nested sparse ConvNets outperform variable-latency solutions naively built assembling single sparse models trained as stand-alone instances, achieving 1) comparable accuracy; 2) remarkable storage savings; and 3) high performance. Moreover, when compared to state-of-the-art dynamic strategies, such as dynamic pruning and layer width scaling, nested sparse ConvNets turn out to be Pareto optimal in the accuracy versus latency space.