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Related to drone out: pass along, at first
To say something in a monotone. A noun or pronoun can be used between "drone" and "out." If you keep droning out information, you're going to put your audience to sleep.
drone (something) out
to make a loud and low-pitched noise; to say something in a low-pitched and monotonous manner. The announcer droned the winning numbers out. She droned out the winning numbers.