How do the early church fathers expound upon the “exaltation of Christ” (Philippians 2:9-11) in light of the immutability of Jesus Christ?
The early church fathers uniformly believed that the pre-existent Word , by being the only-begotten from the Father, was and is fully God in essence (nature).
“We acknowledge a God, and a Son (His Logos), and a Holy Spirit.
These are united in essence-the Father, the Son , and the Spirit.
Now, the Son is the Intelligence, Reason and Wisdom of the Father. And
the Spirit is an emanation, as light from a fire.” – Athenogoras (c.
[Quoting John 1:1] "'...and the Word was God,' of course, for that
which is begotten of God is God." (Against Heresies, Book I, ch. 8,
"The Word, that is, the Son, was always with the Father." (Against
Heresies, Book IV, ch. 20, section 3)
Irenaeus (130-202 C.E.) believed that the Word has added another nature to himself, namely, the human nature. This nature of Jesus Christ is what he considered mutable and passible. When it comes to his divine nature, he is immutable and impassible.
"When the Word became flesh , the immutable became mutable and the
impassible became passible."
Irenaeus (Against Heresis 3:18.3).
Hitherto we have met their irrational conceits with the true conceptions  implied in the Word Son,' as the Lord Himself has
given us. But it will be well next to cite the divine oracles, that
the unalterableness of the Son and His unchangeable nature, which is
the Father's, as well as their perverseness, may be still more fully
proved. The Apostle then, writing to the Philippians, says, Have
this mind in you, which was also in Christ Jesus; who, being in the
form of God, thought it not a prize to be equal with God; but emptied
Himself, taking the form of a servant, being made in the likeness of
men. And, being found in fashion as a man, He humbled Himself,
becoming obedient to death, even the death of the cross. Wherefore God
also highly exalted Him, and gave Him a Name which is above every
name; that in the Name of Jesus every knee should bow, of things in
heaven, and things in earth, and things under the earth, and that
every tongue should confess that Jesus Christ is Lord, to the glory of
God the Father .' Can anything be plainer and more express than
this? He was not from a lower state promoted: but rather, existing as
God, He took the form of a servant, and in taking it, was not promoted
but humbled Himself. Where then is there here any reward of virtue, or
what advancement and promotion in humiliation? For if, being God, He
became man, and descending from on high He is still said to be
exalted, where is He exalted, being God? this withal being plain,
that, since God is highest of all, His Word must necessarily be
highest also. Where then could He be exalted higher, who is in the
Father and like the Father in all things ? Therefore He is beyond
the need of any addition; nor is such as the Arians think Him. For
though the Word has descended in order to be exalted, and so it is
written, yet what need was there that He should humble Himself, as if
to seek that which He had already? And what grace did He receive who
is the Giver of grace ? or how did He receive that Name for worship,
who is always worshipped by His Name? Nay, certainly before He became
man, the sacred writers invoke Him, Save me, O God, for Thy Name's
sake ;'and again, Some put their trust in chariots, and some in
horses, but we will remember the Name of the Lord our .' And while He
was worshipped by the Patriarchs, concerning the Angels it is written,
Let all the Angels of God worship Him .'
And if, as David says in the 71st Psalm, His Name remaineth before the sun, and before the moon, from one generation to another ,' how
did He receive what He had always, even before He now received it? or
how is He exalted, being before His exaltation the Most High? or how
did He receive the right of being worshipped, who before He now
received it, was ever worshipped? It is not a dark saying but a divine
mystery . In the beginning was the Word, and the Word was with God,
and the Word was God;' but for our sakes afterwards the Word was made
flesh.' And the term in question, highly exalted,' does not signify
that the essence of the Word was exalted, for He was ever and is equal
to God ,' but the exaltation is of the manhood. Accordingly this is
not said before the Word became flesh; that it might be plain that
humbled' and exalted' are spoken of His human nature; for where
there is humble estate, there too may be exaltation; and if because of
His taking flesh humbled' is written, it is clear that highly exalted'
is also said because of it. For of this was man's nature in want,
because of the humble estate of the flesh and of death. Since then the
Word, being the Image of the Father and immortal, took the form of the
servant, and as man underwent for us death in His flesh, that thereby
He might offer Himself for us through death to the Father; therefore
also, as man, He is said because of us and for us to be highly
exalted, that as by His death we all died in Christ, so again in the
Christ Himself we might be highly exalted, being raised from the dead,
and ascending into heaven, whither the forerunner Jesus is for us
entered, not into the figures of the true, but into heaven itself, now
to appear in the presence of God for us (Athanasius, Chapter
Athanasius (296-373 C.E.) believed that Jesus Christ, being God himself, was already exalted but because of his exinanition , he took on human nature and was therefore, became lowly in status. His humiliation was his mission. The completion of his work prompted the Father to give back everything he had temporarily given up. In other words, Jesus did not lose his divine nature at his incarnation. He was fully God when he was on earth. However, the use of his unlimited divine powers has become limited due to his own choice and this is what his "self-emptying" really means.
The church fathers interpret the exaltation of Christ in Philippians 2:9-11 in light of his human nature.