In Matthew 2 Herod is seeking to find the birthplace of the Messiah, and the chief priests and scribes correctly identify the birthplace as Bethlehem, citing Micah 5:2, 4. This reveals that even before Christ this passage was understood as messianic.
However, Jesus will condemn the scribes and Pharisees for not having heeded Micah’s teaching. When Jesus says, “For you tithe mint and dill and cumin, and have neglected the weightier matters of the law: justice and mercy and faithfulness” (Mt 23:23) he may be alluding to Micah 6:8, which elevates justice, steadfast love (often rendered mercy in the LXX), and walking humbly with God over the rituals of the law.[1]
Finally, the great commission in Matthew 28, in which Jesus commissions his disciples to make disciples of the nations anticipates the millennial fulfillment of Micah 4:2 in which the nations stream to Jerusalem to learn the ways of the God of Jacob.
[1] Dempster, THOTC, 211.