This paper presents a novel streaming automatic speech recognition (ASR) framework for multi-talker overlapping speech captured by a distant microphone array with an arbitrary geometry. Our framework, named t-SOT-VA, capitalizes on independently developed two recent technologies; array-geometry-agnostic continuous speech separation, or VarArray, and streaming multi-talker ASR based on token-level serialized output training (t-SOT). To combine the best of both technologies, we newly design a t-SOT-based ASR model that generates a serialized multi-talker transcription based on two separated speech signals from VarArray. We also propose a pre-training scheme for such an ASR model where we simulate VarArray's output signals based on monaural single-talker ASR training data. Conversation transcription experiments using the AMI meeting corpus show that the system based on the proposed framework significantly outperforms conventional ones. Our system achieves the state-of-the-art word error rates of 13.7% and 15.5% for the AMI development and evaluation sets, respectively, in the multiple-distant-microphone setting while retaining the streaming inference capability.