IMO, the _delete_by_query page should mention that if an index is moved out of 'hot' phase by ILM, the delete will fail with "FORBIDDEN/8/index write (api)".
I believe this will depend on how your ILM policy is defined. You have the option to perform a couple of actions that will make the index read-only, but if those steps are avoided it should still be possible to update indices on other tiers.
What does your policy look like?