Although end-to-end (E2E) automatic speech recognition (ASR) has shown state-of-the-art recognition accuracy, it tends to be implicitly biased towards the training data distribution which can degrade generalisation. This paper proposes a label-synchronous neural transducer (LS-Transducer), which provides a natural approach to domain adaptation based on text-only data. The LS-Transducer extracts a label-level encoder representation before combining it with the prediction network output. Since blank tokens are no longer needed, the prediction network performs as a standard language model, which can be easily adapted using text-only data. An Auto-regressive Integrate-and-Fire (AIF) mechanism is proposed to generate the label-level encoder representation while retaining low latency operation that can be used for streaming. In addition, a streaming joint decoding method is designed to improve ASR accuracy while retaining synchronisation with AIF. Experiments show that compared to standard neural transducers, the proposed LS-Transducer gave a 12.9% relative WER reduction (WERR) for intra-domain LibriSpeech data, as well as 21.4% and 24.6% relative WERRs on cross-domain TED-LIUM 2 and AESRC2020 data with an adapted prediction network.