Did the early church teach the person of God the Word changed in the incarnation? According to early Christianity did the person of God the Word change in the incarnation? Is it considered mutability in the person that change?
Did the early church teaches the person of God the Word changed in the incarnation?

St. Irenaeus and St.Athansius said it clearly and it is also cited in the Catechism of the Catholic Church.


  460 The Word became flesh to make us "partakers of the divine nature":78 "For this is why the Word became man, and the Son of God became the Son of man: so that man, by entering into communion with the Word and thus receiving divine sonship, might become a son of God."79 "For the Son of God became man so that we might become God."80 "The only-begotten Son of God, wanting to make us sharers in his divinity, assumed our nature, so that he, made man, might make men gods."81
  
  79 St. Irenaeus, Adv. haeres. 3, 19, 1: PG 7/1, 939.
  
  80 St. Athanasius, De inc. 54, 3: PG 25, 192B.


According to early Christianity did the person of God the Word change in the incarnation?

The Word or Logos the Second Person of the Holy Trinity existed in eternity as a "Spirit", and St.John stated in the gospel what is the changes that occured;


  John 1:14 New American Bible (Revised Edition) (NABRE)
  
  And the Word became flesh[a]
      and made his dwelling among us,
      and we saw his glory,
      the glory as of the Father’s only Son,
      full of grace and truth.
  
  Footnotes:
  1:14 Flesh: the whole person, used probably against docetic tendencies (cf. 1 Jn 4:2; 2 Jn 7). Made his dwelling: literally, “pitched his tent/tabernacle.” Cf. the tabernacle or tent of meeting that was the place of God’s presence among his people (Ex 25:8–9). The incarnate Word is the new mode of God’s presence among his people. The Greek verb has the same consonants as the Aramaic word for God’s presence (Shekinah). Glory: God’s visible manifestation of majesty in power, which once filled the tabernacle (Ex 40:34) and the temple (1 Kgs 8:10–11, 27), is now centered in Jesus. Only Son: Greek, monogenēs, but see note on Jn 1:18. Grace and truth: these words may represent two Old Testament terms describing Yahweh in covenant relationship with Israel (cf. Ex 34:6), thus God’s “love” and “fidelity.” The Word shares Yahweh’s covenant qualities.


Is it considered mutability in the person that change?


  "Mutable"adjective
  
  liable or subject to change or alteration.
  
  given to changing; constantly changing; fickle or inconstant:


If you refer "mutability" in the Divine Nature or Essence of God the Son, then it is clearly wrong, as God the Son being "co-substantial" with the Father is immutable.

But, if you refer the "mutability" to the created human soul of Christ the Second Person of the Trinity, then the word "mutability" can be relate to the word "assumed human nature", because the Person of Jesus Christ was change from solely "Spirit or Divine" into a "God-Man", fully God and fully man.


  III. TRUE GOD AND TRUE MAN
  
  464 The unique and altogether singular event of the Incarnation of the Son of God does not mean that Jesus Christ is part God and part man, nor does it imply that he is the result of a confused mixture of the divine and the human. He became truly man while remaining truly God. Jesus Christ is true God and true man.
  
  During the first centuries, the Church had to defend and clarify this truth of faith against the heresies that falsified it.
  
  465 The first heresies denied not so much Christ's divinity as his true humanity (Gnostic Docetism). From apostolic times the Christian faith has insisted on the true incarnation of God's Son "come in the flesh".87 But already in the third century, the Church in a council at Antioch had to affirm against Paul of Samosata that Jesus Christ is Son of God by nature and not by adoption. The first ecumenical council of Nicaea in 325 confessed in its Creed that the Son of God is "begotten, not made, of the same substance (homoousios) as the Father", and condemned Arius, who had affirmed that the Son of God "came to be from things that were not" and that he was "from another substance" than that of the Father.88