In a way that is tender, gentle, and affectionate; with soft, loving touches.
From 'caressing' (present participle of 'caress') plus the adverbial suffix '-ly'. The '-ing' form makes an adjective describing the manner of action.
Adverbs ending in '-ingly' from '-ing' forms are actually redundant (we could just say 'caressingly' is built on top of 'caressingly'), but poets love them for their rolling, musical quality—try saying it aloud and you'll feel what it means.
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