Is it farther or further?
"to feed," Old English grasian "to feed on grass," from græs "grass" (see grass). Cf. Middle Dutch, Middle High German grasen, Dutch grazen, German grasen. Figurative use by 1570s. Related: Grazed; grazing.
"to touch," c.1600, perhaps a transferred sense from graze (v.1) via a notion of cropping grass right down to the ground (cf. German grasen "to feed on grass," used in military sense in reference to cannonballs that rebound off the ground). Related: Grazed; grazing. As a noun from 1690s.
To eat small amounts often: ''I don't eat meals,'' she said. ''I graze all day long''/ Cindy Crawford grazing at the salad bar (1980s+)