For both: A drover, stockie, herdsman, herder.

For sheep: shepherd, drover, sheep-herder, stockie

For cattle: cowboy, cowpuncher, cowpoke, herdsman, drover, stockie

In the UK they are 'drovers' or 'herdsman'. They are usually on foot ,with trained dogs to control and aid movement of the herd.

Specifically, in the USA they are 'Cowboys' and are on horseback/ride horses. They may have dogs too!!!!

cowhand, drover, rancher, stockman, cattleman, herdsman, gaucho

cowhand, drover, rancher, stockman, cattleman, herdsman, gaucho

rancher, bronco, cowhand, cowboy, herdsman, wrangler...