While large language models (LLMs) have been applied to automatic speech recognition (ASR), the task of making the model streamable remains a challenge. This paper proposes a novel model architecture, Transducer-Llama, that integrates LLMs into a Factorized Transducer (FT) model, naturally enabling streaming capabilities. Furthermore, given that the large vocabulary of LLMs can cause data sparsity issue and increased training costs for spoken language systems, this paper introduces an efficient vocabulary adaptation technique to align LLMs with speech system vocabularies. The results show that directly optimizing the FT model with a strong pre-trained LLM-based predictor using the RNN-T loss yields some but limited improvements over a smaller pre-trained LM predictor. Therefore, this paper proposes a weak-to-strong LM swap strategy, using a weak LM predictor during RNN-T loss training and then replacing it with a strong LLM. After LM replacement, the minimum word error rate (MWER) loss is employed to finetune the integration of the LLM predictor with the Transducer-Llama model. Experiments on the LibriSpeech and large-scale multi-lingual LibriSpeech corpora show that the proposed streaming Transducer-Llama approach gave a 17% relative WER reduction (WERR) over a strong FT baseline and a 32% WERR over an RNN-T baseline.