Answer: These flowers are generally considered to be pest free however I did find a report of the sunflower moth larvae that will destroy the flowers of Echinacea (as well as more commonly on sunflowers.) It sounds like the pest, based on your description. Here is the report and photos to see if it is the same pest, Homoeosoma electellum:
Control can be difficult because once the larvae are inside the flower then contact insecticides will not reach them; if you spray the flowers there will be only limited windows when the spray would be effective. Picking off the flowers would prevent the larvae from reaching maturity, but would ruin the garden display. I would suggest you work with your local county extension to verify that this is the correct pest and then based on knowing that, decide how to treat them. If a chemical control is available for home use, they will have the most up to date information on what to use and how/when is best to apply it for maximum results.
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