Keyword spotting (KWS) is essential for voice-driven applications, demanding both accuracy and efficiency. Traditional ASR-based KWS methods, such as greedy and beam search, explore the entire search space without explicitly prioritizing keyword detection, often leading to suboptimal performance. In this paper, we propose an effective keyword-specific KWS framework by introducing a streaming-oriented CTC-Transducer-combined frame-asynchronous system with multi-head frame-asynchronous decoding (MFA-KWS). Specifically, MFA-KWS employs keyword-specific phone-synchronous decoding for CTC and replaces conventional RNN-T with Token-and-Duration Transducer to enhance both performance and efficiency. Furthermore, we explore various score fusion strategies, including single-frame-based and consistency-based methods. Extensive experiments demonstrate the superior performance of MFA-KWS, which achieves state-of-the-art results on both fixed keyword and arbitrary keywords datasets, such as Snips, MobvoiHotwords, and LibriKWS-20, while exhibiting strong robustness in noisy environments. Among fusion strategies, the consistency-based CDC-Last method delivers the best performance. Additionally, MFA-KWS achieves a 47% to 63% speed-up over the frame-synchronous baselines across various datasets. Extensive experimental results confirm that MFA-KWS is an effective and efficient KWS framework, making it well-suited for on-device deployment.