By-the-way definitions

(conjunctive, speech act, idiomatic) Incidentally; a parenthetical statement not timely, central, or crucial to the topic at hand; foregone, passed by, something that has already happened.

His mother will be coming for dinner tomorrow, and, by the way, she volunteered to bring dessert.

prepositional phrase
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(UK, idiomatic) Irrelevantly, off-topic.
adverb
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