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?
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
Did the person of God the Word change in the incarnation? In St. John of Damascene discourse on incarnation, the saint explains that person of God the Word was simple prior to incarnation and becoming composite person at incarnation by hypostatically united two natures in one person.
We say, then, that the divine Person of God the Word exists before all things timelessly and eternally, simple and uncompounded, uncreated, incorporeal, invisible, intangible, and uncircumscribed. ... But, at the same time, we say that in latter times, without leaving the bosom of the Father, the Word came to dwell uncircumscribed in the womb of the holy Virgin, without seed and without being contained, but after a manner known to Him, and in the very same Person as exists before the ages He made flesh subsist for Himself from the holy Virgin. ... And so, He was made flesh and took from her the first-fruits of our clay, a body animated by a rational and intellectual soul, so that the very Person of God the Word was accounted to the flesh. And the Person of the Word which formerly had been simple was made composite.
St. John of Damascene, An Exact Exposition of the Orthodox Faith, Book 3, Chapter 7. (emphasis mine)
In another book, he continued.
Before the Incarnation, the Person of God the Word was simple and uncompounded, bodiless and uncreated. But when it had assumed flesh, it became person to the flesh also, and it became compounded of the divinity, which it always had, and the flesh, which it took on in addition. Being thus found in two natures, it bears the properties of the two, so that the same one person is at once uncreated in its divinity and created in its humanity, both visible and invisible. Otherwise, we are obliged either to divide the one Christ and say that there are two persons, or to deny the difference of the natures and thus introduce change and mingling.
St. John of Damascene, An Exact Exposition of the Orthodox Faith, Book 4, Chapter 5. (emphasis mine)
This view was defined as a dogma three centuries earlier during the Fourth Ecumenical Council at Chalcedon in 451 when the Chalcedonian bishops voted to use the language of one composite person with a Greek word miahypostasis (μιαὑπόστασις). The person of God the Word was becoming composite by assuming humanity, capable to operate humanely and to will what human wills.
Is it considered mutability in the person that change? This change in the person is not a mutability of divine nature. The entirety of Trinity did not become humans but only one of three, the second person becoming man. A divine nature cannot change because of its timelessness property. The change that the saint discussed is about kenotic theology. Whereby a divine person without alteration in His divine nature added an extra nature in time and by doing so relatively at a point in time does change from simple having one nature to composite by having second nature.