  An $\alpha,\beta$-Kempe swap in a properly colored graph interchanges the
colors on some component of the subgraph induced by colors $\alpha$ and
$\beta$. Two $k$-colorings of a graph are $k$-Kempe equivalent if we can form
one from the other by a sequence of Kempe swaps (never using more than $k$
colors). Las Vergnas and Meyniel showed that if a graph is $(k-1)$-degenerate,
then each pair of its $k$-colorings are $k$-Kempe equivalent. Mohar conjectured
the same conclusion for connected $k$-regular graphs. This was proved for $k=3$
by Feghali, Johnson, and Paulusma (with a single exception $K_2\square K_3$,
also called the 3-prism) and for $k\ge 4$ by Bonamy, Bousquet, Feghali, and
Johnson.
  In this paper we prove an analogous result for list-coloring. For a
list-assignment $L$ and an $L$-coloring $\varphi$, a Kempe swap is called
$L$-valid for $\varphi$ if performing the Kempe swap yields another
$L$-coloring. Two $L$-colorings are called $L$-equivalent if we can form one
from the other by a sequence of $L$-valid Kempe swaps. Let $G$ be a connected
$k$-regular graph with $k\ge 3$. We prove that if $L$ is a $k$-assignment, then
all $L$-colorings are $L$-equivalent (again with a single exception $K_2
\square K_3$). When $k\ge 4$, the proof is completely self-contained, so
implies an alternate proof of the result of Bonamy et al.
  Our proofs rely on the following key lemma, which may be of independent
interest. Let $H$ be a graph such that for every degree-assignment $L_H$ all
$L_H$-colorings are $L_H$-equivalent. If $G$ is a connected graph that contains
$H$ as an induced subgraph, then for every degree-assignment $L_G$ for $G$ all
$L_G$-colorings are $L_G$-equivalent.
