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This paper describes Edinburgh’s submissions to the IWSLT2021 multilingual speech translation (ST) task. We aim at improving multilingual translation and zero-shot performance in the constrained setting (without using any extra training data) through methods that encourage transfer learning and larger capacity modeling with advanced neural components. We build our end-to-end multilingual ST model based on Transformer, integrating techniques including adaptive speech feature selection, language-specific modeling, multi-task learning, deep and big Transformer, sparsified linear attention and root mean square layer normalization. We adopt data augmentation using machine translation models for ST which converts the zero-shot problem into a zero-resource one. Experimental results show that these methods deliver substantial improvements, surpassing the official baseline by > 15 average BLEU and outperforming our cascading system by > 2 average BLEU. Our final submission achieves competitive performance (runner up).