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Deep learning-based algorithms have greatly improved the performance of remote sensing image (RSI) super-resolution (SR). However, increasing network depth and parameters cause a huge burden of computing and storage. Directly reducing the depth or width of existing models results in a large performance drop. We observe that the SR difficulty of different regions in an RSI varies greatly, and existing methods use the same deep network to process all regions in an image, resulting in a waste of computing resources. In addition, existing SR methods generally predefine integer scale factors and cannot perform stepless SR, i.e., a single model can deal with any potential scale factor. Retraining the model on each scale factor wastes considerable computing resources and model storage space. To address the above problems, we propose a saliency-aware dynamic routing network (SalDRN) for lightweight and stepless SR of RSIs. First, we introduce visual saliency as an indicator of region-level SR difficulty and integrate a lightweight saliency detector into the SalDRN to capture pixel-level visual characteristics. Then, we devise a saliency-aware dynamic routing strategy that employs path selection switches to adaptively select feature extraction paths of appropriate depth according to the SR difficulty of sub-image patches. Finally, we propose a novel lightweight stepless upsampling module whose core is an implicit feature function for realizing mapping from low-resolution feature space to high-resolution feature space. Comprehensive experiments verify that the SalDRN can achieve a good trade-off between performance and complexity. The code is available at \url{https://github.com/hanlinwu/SalDRN}.