Any radical transformation in case of incarnation without fall?
The answer is Yes according to Eastern Church Fathers as they teaches the word Theosis.
Theosis, or divinization ("divinization may also refer to apotheosis, lit. "making divine"), is a transformative process whose aim is likeness to or union with God, as taught by the Eastern Orthodox Church and Eastern Catholic Churches. As a process of transformation, theosis is brought about by the effects of catharsis (purification of mind and body) and theoria ('illumination' with the 'vision' of God). According to Eastern Christian teaching, theosis is very much the purpose of human life. It is considered achievable only through a synergy (or cooperation) between human activity and God's uncreated energies (or operations).https://en.wikipedia.org/wiki/Theosis_(Eastern_Christian_theology)
Adam & Eve have access to the Tree of Life, and the fruit of the Tree of Life is what the Catholic Church believe today as the Holy Eucharist. The transformation, the Catholic Church teaches is a "transfigured bodies" for those who lived a Eucharistic Life.
Saint Albert the Great taught that the Eucharist, the Body and Blood of Christ, is the Fruit of the Tree of Life.https://en.wikipedia.org/wiki/Tree_of_life
1000 This "how" exceeds our imagination and understanding; it is accessible only to faith. Yet our participation in the Eucharist already gives us a foretaste of Christ's transfiguration of our bodies:
Just as bread that comes from the earth, after God's blessing has been invoked upon it, is no longer ordinary bread, but Eucharist, formed of two things, the one earthly and the other heavenly: so too our bodies, which partake of the Eucharist, are no longer corruptible, but possess the hope of resurrection.556
Question: How it would be possible to lay down one's life for one's friends if there is no death?
If Adam & Eve did not Fall then Jesus would still come as Creation & Incarnation is not a separate plan but a complete plan of God based on the teaching of Blessed John Duns Scotus on the Primacy of Christ. According to Blessed Scotus Jesus will still come not as a Redeemer but King of His creation.
The theology of John Duns Scotus places Christ at the centre of a universe ordered by love. Christ is presented as the basis of all nature, grace and glory – the most perfect model of humanity. He is at the beginning, the centre and the end of the universe.
St. Paul tells us that Christ was the first-born of all creation, and Scotus’ theology makes sense of this affirmation. Scotus did not believe that the acts of creation and Incarnation were separate, but part of one divine plan. So rather than the Incarnation being a sort of “Plan B” to rescue humanity after the fall, in Scotus’ theology it is the whole purpose of creation. Christ is the masterpiece of love in the midst of a creation designed for love, rather than a divine plumber come to fix the mess of original sin. Thus the Incarnation is placed by Scotus in the context of creation and not of human sin.
The Primacy of Christ in John Duns Scotus: An Assessment.
Note: I heard somewhere that there would be some transition like this, just without sorrow and pain. Similar to Mary's assumption. How it would look like? Would the person after the transition still be on the Earth?
This is a different scenario on your premise above wherein Adam Eve did not Fall. Mary's Assumptions happened because original sin enters the world. You might probably asking on the appearance of the faithful souls in the New Paradise. The Book of Revelation speaks of a Eucharistic Era, so based on Catholic Church Catechism#1000 souls will be given a foretaste of Christ "transfigured bodies".
In closing,the Primacy of Christ teaches that the Omnipotent God in all eternity knew that Adam & Eve will fall into temptation even the Fall of Lucifer as the Wisdom of God knew everything.
"For the spirit of the Lord fills the world,
is all-embracing, and knows whatever is said.(Wisdom1:7)
The Primacy of Christ is the ultimate radical transformation, A
Creator transforming into a creature assuming humanity devoid of all powers majesty, but why?
St.Athansius has the answer;
"God becomes man, so that man shall become gods."(St.Athanasius)
Jesus the Begotten Son of God has to come down to show us the Way how to enter His Father's Kingdom, and there's is only one way and the first one who teach the Apostles the Way that Jesus had shown by His life is the Theotokos. The Catholic Church teaches that the Way to Heaven and "transformation" is the Way of the Cross.
As St.Paul boldly exclaimed his own transformation;
"I have been crucified with Christ, and I no longer live, but Christ lives in me. The life I live in the body, I live by faith in the Son of God, who loved me and gave Himself up for me. (Galatians2:20)