Cross-domain Sequential Recommendation (CSR) is an emerging yet challenging task that depicts the evolution of behavior patterns for overlapped users by modeling their interactions from multiple domains. Existing studies on CSR mainly focus on using composite or in-depth structures that achieve significant improvement in accuracy but bring a huge burden to the model training. Moreover, to learn the user-specific sequence representations, existing works usually adopt the global relevance weighting strategy (e.g., self-attention mechanism), which has quadratic computational complexity. In this work, we introduce a lightweight external attention-enhanced GCN-based framework to solve the above challenges, namely LEA-GCN. Specifically, by only keeping the neighborhood aggregation component and using the Single-Layer Aggregating Protocol (SLAP), our lightweight GCN encoder performs more efficiently to capture the collaborative filtering signals of the items from both domains. To further alleviate the framework structure and aggregate the user-specific sequential pattern, we devise a novel dual-channel External Attention (EA) component, which calculates the correlation among all items via a lightweight linear structure. Extensive experiments are conducted on two real-world datasets, demonstrating that LEA-GCN requires a smaller volume and less training time without affecting the accuracy compared with several state-of-the-art methods.