  We give a natural definition of rowmotion for $321$-avoiding permutations, by
translating, through bijections involving Dyck paths and the Lalanne--Kreweras
involution, the analogous notion for antichains of the positive root poset of
type $A$. We prove that some permutation statistics, such as the number of
fixed points, are homomesic under rowmotion, meaning that they have a constant
average over its orbits.
  Our setting also provides a more natural description of the celebrated
Armstrong--Stump--Thomas equivariant bijection between antichains and
non-crossing matchings in types $A$ and $B$, by showing that it is equivalent
to the Robinson--Schensted--Knuth correspondence on $321$-avoiding permutations
permutations.
