Restoring 27 databases; they all restored properly, but 15 of them gave a warning along these lines:
Warning: Wordbreaker, filter, or protocol handler used by catalog ‘FOOBARBAZ’ does not exist on this instance. Use sp_help_fulltext_catalog_components and sp_help_fulltext_system_components check for mismatching components. Rebuild catalog is recommended.
Read on for the solution.