Summary: Suppose a finite set \(X\) is repeatedly transformed by a sequence of permutations of a certain type acting on an initial element \(x\) to produce a final state \(y\). For example, in genomics applications, \(X\) could be a set of genomes and the permutations certain genome ‘rearrangements’ or, in group theory, \(X\) could be the set of configurations of a Rubik’s cube and the permutations certain specified moves. We investigate how ‘different’ the resulting state \(y'\) to \(y\) can be if a slight change is made to the sequence, either by deleting one permutation, or replacing it with another. Here the ‘difference’ between \(y\) and \(y'\) might be measured by the minimum number of permutations of the permitted type required to transform \(y\) to \(y'\), or by some other metric. We discuss this first in the general setting of sensitivity to perturbation of walks in Cayley graphs of groups with a specified set of generators. We then investigate some permutation groups and generators arising in computational genomics, and the statistical implications of the findings.