The Transformer self-Attention network has recently shown promising performance as an alternative to recurrent neural networks (RNNs) in end-To-end (E2E) automatic speech recognition (ASR) systems. However, the Transformer has a drawback in that the entire input sequence is required to compute self-Attention. In this paper, we propose a new block processing method for the Transformer encoder by introducing a context-Aware inheritance mechanism. An additional context embedding vector handed over from the previously processed block helps to encode not only local acoustic information but also global linguistic, channel, and speaker attributes. We introduce a novel mask technique to implement the context inheritance to train the model efficiently. Evaluations of the Wall Street Journal (WSJ), Librispeech, VoxForge Italian, and AISHELL-1 Mandarin speech recognition datasets show that our proposed contextual block processing method outperforms naive block processing consistently. Furthermore, the attention weight tendency of each layer is analyzed to clarify how the added contextual inheritance mechanism models the global information.