Sequential decoding, commonly applied to substitution channels, is a sub-optimal alternative to Viterbi decoding with significantly reduced memory costs. In this work, a sequential decoder for convolutional codes over channels that are prone to insertion, deletion, and substitution errors, is described and analyzed. Our decoder expands the code trellis by a new channel-state variable, called drift state, as proposed by Davey and MacKay. A suitable decoding metric on that trellis for sequential decoding is derived, generalizing the original Fano metric. The decoder is also extended to facilitate the simultaneous decoding of multiple received sequences that arise from a single transmitted sequence. Under low-noise environments, our decoding approach reduces the decoding complexity by a couple orders of magnitude in comparison to Viterbi's algorithm, albeit at slightly higher bit error rates. An analytical method to determine the computational cutoff rate is also suggested. This analysis is supported with numerical evaluations of bit error rates and computational complexity, which are compared with respect to optimal Viterbi decoding.