For a list assignment $L$ and an $L$-coloring $\varphi$, a Kempe swap in $\varphi$ is \emph{$L$-valid} if it yields another $L$-coloring. Two $L$-colorings are \emph{$L$-equivalent} if we can form one from another by a sequence of $L$-valid Kempe swaps. And a graph $G$ is \emph{$L$-swappable} if every two of its $L$-colorings are $L$-equivalent. We consider $L$-swappability of line graphs of planar graphs with large maximum degree. Let $G$ be a planar graph with $Δ(G)\ge 9$ and let $H$ be the line graph of $G$. If $L$ is a $(Δ(G)+1)$-assignment to $H$, then $H$ is $L$-swappable. Let $G$ be a planar graph with $Δ(G)\ge 15$ and let $H$ be the line graph of $G$. If $L$ is a $Δ(G)$-assignment to $H$, then $H$ is $L$-swappable. The first result is analogous to one for $L$-choosability by Borodin, which was later strengthened by Bonamy. The second result is analogous to another for $L$-choosability by Borodin, which was later strengthened by Borodin, Kostochka, and Woodall.