In a grumbling manner; while complaining or expressing discontent in a low voice.
From grumble plus -ing (present participle) plus -ly (adverbial suffix), following the standard English pattern of converting verbs to manner adverbs.
Adverbs ending in -ly that come from complaint-related verbs (grumblingly, complainingly, grudgingly) are wonderfully useful for adding character to dialogue—they instantly show us a character's attitude without explicit description.
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