sway
v.v. to move slowly from side to side. You can also use it to describe someone changing their mind or being influenced by an opinion.
v. to move rhythmically or slowly back and forth or from side to side; figuratively, to exert influence or control over someone's opinions or actions.
The tall trees sway gently in the wind.
The speaker's emotional story began to sway the crowd, changing their minds about the new law.
The ship began to sway as the storm intensified, making it difficult for the crew to maintain their footing on the slick wooden deck.
From earlier swey (“to fall, swoon”), from Middle English sweyen, from Old English swǣġan (“to bend, bow”), from Proto-West Germanic swaigijan, from Proto-Germanic swaigijaną, from Proto-Indo-European sweh₁- See also Saterland Frisian swooie (“to swing, wave, wobble”); also Lithuanian svai̇̃gti (“to become giddy or dizzy”), the second element of Avestan 𐬞𐬀𐬌𐬭𐬌-𐬱𐬑𐬎𐬀𐬑𐬙𐬀 (paⁱri-šxuaxta, “to surround”), Sanskrit स्वजते (svájate, “he embraces, enfolds”). The noun derived from the verb.
The verb is both transitive (to influence someone) and intransitive (to move back and forth).