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Convolutional neural networks have witnessed remarkable improvements in computational efficiency in recent years. A key driving force has been the idea of trading-off model expressivity and efficiency through a combination of $1\times 1$ and depth-wise separable convolutions in lieu of a standard convolutional layer. The price of the efficiency, however, is the sub-optimal flow of information across space and channels in the network. To overcome this limitation, we present MUXConv, a layer that is designed to increase the flow of information by progressively multiplexing channel and spatial information in the network, while mitigating computational complexity. Furthermore, to demonstrate the effectiveness of MUXConv, we integrate it within an efficient multi-objective evolutionary algorithm to search for the optimal model hyper-parameters while simultaneously optimizing accuracy, compactness, and computational efficiency. On ImageNet, the resulting models, dubbed MUXNets, match the performance (75.3% top-1 accuracy) and multiply-add operations (218M) of MobileNetV3 while being 1.6$\times$ more compact, and outperform other mobile models in all the three criteria. MUXNet also performs well under transfer learning and when adapted to object detection. On the ChestX-Ray 14 benchmark, its accuracy is comparable to the state-of-the-art while being $3.3\times$ more compact and $14\times$ more efficient. Similarly, detection on PASCAL VOC 2007 is 1.2% more accurate, 28% faster and 6% more compact compared to MobileNetV2. Code is available from https://github.com/human-analysis/MUXConv