  An extension of an induced path $P$ in a graph $G$ is an induced path $P'$
such that deleting the endpoints of $P'$ results in $P$. An induced path in a
graph is said to be avoidable if each of its extensions is contained in an
induced cycle. In 2019, Beisegel, Chudovsky, Gurvich, Milani\v{c}, and
Servatius conjectured that every graph that contains an induced $k$-vertex path
also contains an avoidable induced path of the same length, and proved the
result for $k = 2$. The case $k = 1$ was known much earlier, due to a work of
Ohtsuki, Cheung, and Fujisawa in 1976. The conjecture was proved for all $k$ in
2020 by Bonamy, Defrain, Hatzel, and Thiebaut. In the present paper, using a
similar approach, we strengthen their result from a reconfiguration point of
view. Namely, we show that in every graph, each induced path can be transformed
to an avoidable one by a sequence of shifts, where two induced $k$-vertex paths
are shifts of each other if their union is an induced path with $k+1$ vertices.
We also obtain analogous results for not necessarily induced paths and for
walks. In contrast, the statement cannot be extended to trails or to isometric
paths.
