Continuous sign language recognition (cSLR) is a public significant task that transcribes a sign language video into an ordered gloss sequence. It is important to capture the fine-grained gloss-level details, since there is no explicit alignment between sign video frames and the corresponding glosses. Among the past works, one promising way is to adopt a one-dimensional convolutional network (1D-CNN) to temporally fuse the sequential frames. However, CNNs are agnostic to similarity or dissimilarity, and thus are unable to capture local consistent semantics within temporally neighboring frames. To address the issue, we propose to adaptively fuse local features via temporal similarity for this task. Specifically, we devise a Multi-scale Local-Temporal Similarity Fusion Network (mLTSF-Net) as follows: 1) In terms of a specific video frame, we firstly select its similar neighbours with multi-scale receptive regions to accommodate different lengths of glosses. 2) To ensure temporal consistency, we then use position-aware convolution to temporally convolve each scale of selected frames. 3) To obtain a local-temporally enhanced frame-wise representation, we finally fuse the results of different scales using a content-dependent aggregator. We train our model in an end-to-end fashion, and the experimental results on RWTH-PHOENIX-Weather 2014 datasets (RWTH) demonstrate that our model achieves competitive performance compared with several state-of-the-art models.