To go toward or proceed to a place or direction; to betake oneself somewhere.
来月、新しい部署へ赴くことになった。
Next month I will be assigned to (and go to) a new department.
彼は会場の奥へ静かに赴いた。
He quietly went toward the back of the venue.
調査は別の方向へ赴き、問題の本質が明らかになった。
The investigation proceeded in a different direction, and the essence of the problem became clear.
To assume a certain state or to turn toward a particular condition; to face or become (a certain way).
会議の雰囲気は次第に実務的な方向へ赴いた。
The mood of the meeting gradually moved in a practical direction.
話が進むにつれて、議論は深刻な方へ赴いていった。
As the discussion went on, it moved toward a more serious tone.
Archaic: to abide by, consent to, or comply with (an older/literary usage not common in modern Japanese).
古い書簡には、主君の命に赴く（したがう）と書かれていることがある。
In old letters you sometimes find wording that one 'abides by' the lord's order.