This paper addresses sound source separation and speech recognition for moving sound sources. Real-world applications such as robots should cope with both moving and stationary sound sources. However, most studies assume only stationary sound sources. We introduce two key techniques to cope with moving sources, that is, Adaptive Step-size control (AS) and Optima Controlled Recursive Average (OCRA) to improve blind source separation. We implemented a real-time robot audition system with these techniques for our humanoid robot ASIMO with an 8ch microphone array by using HARK which is our open-source software for robot audition. The performance of the system will be shown through sound source separation for moving sources and automatic speech recognition of separated speeches.