  Given a multigraph $G$ and a positive integer $t$, the distance-$t$ chromatic
index of $G$ is the least number of colours needed for a colouring of the edges
so that every pair of distinct edges connected by a path of fewer than $t$
edges must receive different colours. Let $\pi'_t(d)$ and $\tau'_t(d)$ be the
largest values of this parameter over the class of planar multigraphs and of
(simple) trees, respectively, of maximum degree $d$. We have that $\pi'_t(d)$
is at most and at least a non-trivial constant multiple larger than
$\tau'_t(d)$. (We conjecture $\limsup_{d\to\infty}\pi'_2(d)/\tau'_2(d) =9/4$ in
particular.) We prove for odd $t$ the existence of a quantity $g$ depending
only on $t$ such that the distance-$t$ chromatic index of any planar multigraph
of maximum degree $d$ and girth at least $g$ is at most $\tau'_t(d)$ if $d$ is
sufficiently large. Such a quantity does not exist for even $t$. We also show a
related, similar phenomenon for distance vertex-colouring.
