Many text generation tasks are copy-oriented. For instance, nearly 30% content of news summaries is copied. The copy rate is even higher in Grammatical Error Correction (GEC). However, existing generative models generate texts through word-by-word decoding, which may lead to factual inconsistencies and slow inference. While Elementary Discourse Units (EDUs) are outstanding extraction units, EDU-based extractive methods can alleviate the aforementioned problems. As a consequence, we propose EDUCopy, a framework that integrates the behavior of copying EDUs into generative models. The main idea of EDUCopy is to use special index tags to represent the copied EDUs during generation. Specifically, we extract important EDUs from input sequences, finetune generative models to generate sequences with special index tags, and restore the generated special index tags into corresponding text spans. By doing so, EDUCopy reduces the number of generated tokens significantly. To verify the effectiveness of EDUCopy, we conduct experiments on the news summarization datasets CNNDM, NYT and the GEC datasets FCE, WI-LOCNESS. While achieving notable ROUGE and M2 scores, GPT-4 evaluation validates the strength of our models in terms of factual consistency, fluency, and overall performance. Moreover, compared to baseline models, EDUCopy achieves a significant acceleration of 1.65x.