Fix a planar graph $G$ and a list-assignment $L$ with $|L(v)|=10$ for all $v\in V(G)$. Let $α$ and $β$ be $L$-colorings of $G$. A recoloring sequence from $α$ to $β$ is a sequence of $L$-colorings, beginning with $α$ and ending with $β$, such that each successive pair in the sequence differs in the color on a single vertex of $G$. We show that there exists a constant $C$ such that for all choices of $α$ and $β$ there exists a recoloring sequence $σ$ from $α$ to $β$ that recolors each vertex at most $C$ times. In particular, $σ$ has length at most $C|V(G)|$. This confirms a conjecture of Dvořák and Feghali. For our proof, we introduce a new technique for quickly showing that many configurations are reducible. We believe this method may be of independent interest and will have application to other problems in this area.