The kenosis
'He emptied himself' — what is given up: the exercise of divine attributes, status and glory, or nothing of his nature; the two-natures readings.
Philippians 2:5-11
5Have this in your mind, which was also in Christ Jesus,
6who, existing in the form of God, didn’t consider equality with God a thing to be grasped,
7but emptied himself, taking the form of a servant, being made in the likeness of men.
8And being found in human form, he humbled himself, becoming obedient to the point of death, yes, the death of the cross.
9Therefore God also highly exalted him, and gave to him the name which is above every name,
10that at the name of Jesus every knee should bow, of those in heaven, those on earth, and those under the earth,
11and that every tongue should confess that Jesus Christ is Lord, to the glory of God the Father.
What each tradition says
Rabbinic / Midrashic
No sourced position found. Checked The Jewish Encyclopedia (jewishencyclopedia.com): 'Jesus of Nazareth' and 'Saul of Tarsus' never cite Phil 2:5-11, 'Memra'/'Logos' treat the Johannine Word only, and the archive.org volume scans (vol. 7, 1904, letter K) show no 'Kenosis' article; the self-emptying is an inner-Christian dispute the 1905 JE does not treat.
Patristic
Chrysostom says Christ had equality with God by nature, not by seizure, and it could never be parted from him. So he did not lay it aside as usurped but hid it. He emptied himself by taking the form of a servant, being made in the likeness of men, not by constraint.
John Chrysostom (c. 349-407), Homilies on the Epistle to the Philippians, in NPNF ser. 1, vol. 13 (ed. Philip Schaff; T&T Clark, Edinburgh, 1889) — Homily VII on Philippians 2:5-11
Catholic
According to Catholic theology, the abasement of the Word consists in the assumption of humanity and the simultaneous concealment of the Divinity. Christ's abasement is seen first in His subjecting Himself to the laws of human birth and growth and to the lowliness of fallen human nature.
A. J. Maas, The Catholic Encyclopedia, vol. 8, s.v. 'Kenosis' (New York: Robert Appleton Company, 1910) — section 'Catholic teaching'
Orthodox
John of Damascus says that Christ bent the heavens and descended to earth, yet He humbled without humiliation His lofty station, which could not be humbled. He condescends to His servants. His descent, then, is not an actual humbling of that station but a condescension.
John of Damascus (c. 675-749), An Exact Exposition of the Orthodox Faith, in NPNF ser. 2, vol. 9 (trans. S. D. F. Salmond; Edinburgh, 1899) — Book III, ch. 1 ('Concerning the Divine OEconomy...')
Reformed / Protestant
Calvin says Christ could not divest himself of Godhead. He kept it concealed for a time, so it was not seen under the weakness of the flesh. Thus he laid aside his glory in the view of men, not by lessening it but by concealing it.
John Calvin (1509-1564), Commentaries on the Epistles of Paul the Apostle to the Philippians, Colossians, and Thessalonians (trans. John Pringle; Edinburgh: Calvin Translation Society, 1851) — Commentary on Philippians 2:7
Evangelical
Wesley says Christ emptied himself of that divine fullness, which he received again at his exaltation. Though he remained full, he appeared as if empty, veiling his fullness from men and angels. In some sense, he also renounced the glory he had before the world began.
John Wesley (1703-1791), Explanatory Notes upon the New Testament (1755); CCEL's 'Wesley's Notes on the Bible' — Note on Philippians 2:7