'sway' (the root of 'swayer') originates from Old Norse, specifically the word 'sveigja', where the root 'sveig-' meant 'to bend' or 'to curve'.

'sveigja' influenced early Germanic forms and entered Middle English as forms like 'swayen'/'sweien', which eventually became the modern English 'sway' and then the agent noun 'swayer'.

Initially it meant 'to bend or swing'; over time it broadened to mean physical back-and-forth movement and, by extension, the figurative sense 'to influence or control', which led to nouns like 'swayer' meaning someone who exerts such influence.