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Currently, processors will for the most part just run on any fulltext field coming along (or, in the case of the "Ignore case" processor, optionally also on strings). However, since processors have configuration forms, there is really no reason not to just provide a form where the user can select all fields the processor should run on. Probably as a scrollable select list, like for the "Aggregated fulltext fields" data alteration.
Right now, there is e.g. no way to tell the "HTML fitler" to only run on the node body, and not other, non-HTML fields. And I imagine that only in very few cases all fulltext data will be HTML.