Capsule networks (CapsNets) have recently gotten attention as a novel neural architecture. This paper presents the sequential routing framework which we believe is the first method to adapt a CapsNet-only structure to sequence-to-sequence recognition. Input sequences are capsulized then sliced by a window size. Each slice is classified to a label at the corresponding time through iterative routing mechanisms. Afterwards, losses are computed by connectionist temporal classification (CTC). During routing, the required number of parameters can be controlled by the window size regardless of the length of sequences by sharing learnable weights across the slices. We additionally propose a sequential dynamic routing algorithm to replace traditional dynamic routing. The proposed technique can minimize decoding speed degradation caused by the routing iterations since it can operate in a non-iterative manner without dropping accuracy. The method achieves a 1.1% lower word error rate at 16.9% on the Wall Street Journal corpus compared to bidirectional long short-term memory-based CTC networks. On the TIMIT corpus, it attains a 0.7% lower phone error rate at 17.5% compared to convolutional neural network-based CTC networks (Zhang et al., 2016).