  The class of free multiarrangements is known to be closed under taking
localizations. We extend this result to the stronger notions of inductive and
recursive freeness.
  As an application, we prove that recursively free multiarrangements are
compatible with the product construction for multiarrangements. In addition, we
show how our results can be used to derive that some canonical classes of free
multiarrangements are not inductively free.
