How does the revelation of Blessed Emmerich harmonize the teaching of Catholicism and Eastern Orthodoxy in Mary's Assumption?
Before I begin, I will start with a caveat!
The revelations of Blessed Anne Catherine Emmerich are a treasure chest of information that can not be always verified or completely sanctioned by the Church. Because Blessed Catherine Emmerich did not write her own revelations, but were in fact written and edited by Brentano, the Church in her process of beatification would not allow these revelations to be taken as means to support her case. The reason, although doubtful, was that Brentano may have added or altered the revelations in question.
The caveat being said, I personally believe that for the most part these revelations are bang on and historical research and modern archaeology have truly vindicated this caveat all together.
Now let us se how these revelations harmonize and to what degree with the two great Churches of the East and West.
The Assumption of the Virgin Mary was dogmatically defined by Pope Pius XII on 1 November 1950, in the apostolic constitution Munificentissimus Deus by exercising papal infallibility. While the Catholic Church and Eastern Orthodox Church believe in the Dormition of the Mother of God (Dormition of the Theotokos or "the Falling Asleep of the Mother of God"), whether Mary had a physical death has not been dogmatically defined.
Although not an infallible statement Pope John Paul II said on June 25, 1997 that “Mary experienced natural death prior to her assumption into Heaven." See the General Audience of Pope John Paul’s statement: Mary and the human drama of death.
While the Catholic Church speaks of Mary’s Assumption and the Eastern Orthodox speaks of Mary’s Dormitione,; the Churches in question are basically expressing the same faith in slightly different angles so to speak.
Many Catholics believe that Mary first died before being assumed, but they believe that she was miraculously resurrected before being assumed. Others believe she was assumed bodily into Heaven without first dying. Either understanding may be legitimately held by Catholics, with Eastern Catholics observing the Feast as the Dormition.
Many theologians note by way of comparison that in the Catholic Church, the Assumption is dogmatically defined, while in the Eastern Orthodox tradition, the Dormition is less dogmatically than liturgically and mystically defined. Such differences spring from a larger pattern in the two traditions, wherein Catholic teachings are often dogmatically and authoritatively defined – in part because of the more centralized structure of the Catholic Church – while in Eastern Orthodoxy, many doctrines are less authoritative.
Orthodox tradition is clear and unwavering in regard to the central point [of the Dormition]: the Holy Virgin underwent, as did her Son, a physical death, but her body – like His – was afterwards raised from the dead and she was taken up into heaven, in her body as well as in her soul. She has passed beyond death and judgement, and lives wholly in the Age to Come. The Resurrection of the Body ... has in her case been anticipated and is already an accomplished fact. That does not mean, however, that she is dissociated from the rest of humanity and placed in a wholly different category: for we all hope to share one day in that same glory of the Resurrection of the Body which she enjoys even now. - Assumption vs. Dormition
The revelations of Catherine Emmerich are in complete harmony with both Churches on this and adds nothing new except more details in her narration.
This poses a problem for the Eastern Orthodox who maintain Mary died at Jerusalem.
A narrative known as the Euthymiaca Historia (written probably by Cyril of Scythopolis in the 5th century) relates how the Emperor Marcian and his wife, Pulcheria, requested the relics of the Virgin Mary from Juvenal, the Patriarch of Jerusalem, while he was attending the Council of Chalcedon (451). According to the account, Juvenal replied that, on the third day after her burial, Mary's tomb was discovered to be empty, only her shroud being preserved in the church of Gethsemane. In 452 the shroud was sent to Constantinople, where it was kept in the Church of Our Lady of Blachernae (Panagia Blacherniotissa).4
According to other traditions, it was the Cincture of the Virgin Mary which was left behind in the tomb, or dropped by her during Assumption.
The Armenian Patriarchate Armenian Apostolic Church of Jerusalem and Greek Orthodox Church of Jerusalem are in possession of the shrine. The Syriacs, the Copts, and the Ethiopians have minor rights.
A legend, which was first mentioned by Epiphanius of Salamis in the 4th century AD, purported that Mary may have spent the last years of her life in Ephesus. The Ephesians derived it from John's presence in the city, and Jesus’ instructions to John to take care of Mary after his death. Epiphanius, however, pointed out that although the Bible mentions John leaving for Asia, it makes no mention of Mary going with him.5 The Eastern Orthodox Church tradition believes that Virgin Mary lived in the vicinity of Ephesus, where there is a place currently known as the House of the Virgin Mary and venerated by Catholics and Muslims, but argues that she only stayed there for a few years.
The pilgrim Antoninus of Piacenza, writing of travels in 560-570 CE, mentions in that valley was "the basilica of the Blessed Mary, which they say was her house; in which is shown a sepulchre, from which they say that the Blessed Mary was taken up into heaven." Later, Saints Epiphanius of Salamis, Gregory of Tours, Isidore of Seville, Modest, Sophronius of Jerusalem, German of Constantinople, Andrew of Crete, and John of Damascus talk about the tomb being in Jerusalem, and bear witness that this tradition was accepted by all the Churches of Easbt and West. - Tomb of the Virgin Mary

Twelfth-century façade of Mary's Tomb at Jerusalem
In 1881, while using the revelations as described in the book by Brentano based on his conversations with Emmerich, a French priest, Fr. Julien Gouyet discovered the house of Mary at Ephesus.
Discovery in Turkey
On October 18, 1881, relying on the descriptions in the book by Brentano based on his conversations with Emmerich, a French priest, the Abbé Julien Gouyet discovered a small stone building on a mountain overlooking the Aegean Sea and the ruins of ancient Ephesus in Turkey. He believed it was the house described by Emmerich and where the Virgin Mary had lived the final years of her life.
Abbé Gouyet's discovery was not taken seriously by most people, but ten years later, urged by Sister Marie de Mandat-Grancey, DC, two Lazarist missionaries, Father Poulin and Father Jung, from Smyrna rediscovered the building on July 29, 1891, using the same source for a guide. They learned that the four-walled, roofless ruin had been venerated for a long time by members of the mountain village of Şirince, 17 km distant, who were descended from the early Christians of Ephesus. The house is called Panaya Kapulu ("Doorway to the Virgin"). Every year pilgrims made a pilgrimage to the site on August 15, the date on which most of the Christian world celebrated Mary's Dormition/Assumption.
Sister Marie de Mandat-Grancey was named Foundress of Mary's House by the Catholic Church and was responsible for acquiring, restoring and preserving Mary's House and surrounding areas of the mountain from 1891 until her death in 1915.[16] The discovery revived and strengthened a Christian tradition dating from the 12th century, 'the tradition of Ephesus', which has competed with the older 'Jerusalem tradition' about the place of the Blessed Virgin's dormition. Due to the actions of Pope Leo XIII in 1896 and Pope John XXIII in 1961, the Catholic Church first removed plenary indulgences from the Church of the Dormition in Jerusalem and then bestowed them for all time to pilgrims to Mary's House in Ephesus.
Taking away the indulgences from the Church of the Dormition in Jerusalem and bestowing them on Mary’s house in Ephesus, the Catholic Church showed to the world which tradition it not only favoured the Ephesus tradition, but modern popes have shown which tradition they believed to be more authentic.
The Roman Catholic Church has never pronounced on the authenticity of the house, for lack of scientifically acceptable evidence. It has, however, from the blessing of the first pilgrimage by Pope Leo XIII in 1896, taken a positive attitude towards the site. Pope Pius XII, in 1951, following the definition of the dogma of the Assumption in 1950, elevated the house to the status of a Holy Place, a privilege later made permanent by Pope John XXIII. The site is visited and venerated by Muslims as well as Christians. Pilgrims drink from a spring under the house which is believed to have healing properties. A liturgical ceremony is held here every year on August 15 to commemorate the Assumption of Mary.
Papal visits
Pope Paul VI visited the shrine on July 26, 1967, and Pope John Paul II on November 30, 1979. Pope Benedict XVI visited this shrine on November 29, 2006, during his four-day pastoral trip to Turkey. - House of the Virgin Mary
As for the actual tomb of the Virgin Mary the Catholic Encyclopedia states the following:
The tomb of the Blessed Virgin is venerated in the Valley of Cedron, near Jerusalem. Modern writers hold, however, that Mary died and was buried at Ephesus.
Catherine Emmerich’s revelations tell us that Mary lived at Ephesus with St. John, there she died and was placed in a tomb. St. Thomas having missed the funeral rites, asked to see her in the tomb, but she was no longer there!
What is more is that the tomb itself although not far from her house remains unknown. She states in the writings that at a future date the tomb of the Virgin Mary will be discovered...
The Blessed Virgin's Burial
When it was time to bear the coffin to the grotto, one half-hour distant, Peter and John raised it from the litter and carried it in their hands to the door of the house, outside of which it was again laid on the litter, which Peter and John then raised upon their shoulders. Six of the Apostles thus carried it in turn. The coffin hung between the bearers as in a cradle, for the poles of the litter were run through leathern straps, or matting. Some of the Apostles walked before the coffin praying, and after it came the women. Lamps, or lanterns on poles, were carried.
Before reaching the grotto, the litter was set down. Four of the Apostles bore the coffin in, and placed it in the hollow of the tomb. All went, one by one, into the grotto, where they knelt in prayer before the holy body, honoring it and taking leave of it. Then the tomb was shut in by a wicker screen that extended from the front edge of the tomb to the top of the vaulted wall above. Before the entrance of the grotto they made a trench, which they planted so thickly with blooming flowers and bushes covered with berries that one could gain access to it only from the side, and that only by making his way through the underwood.
On the night following the burial took place the bodily Assumption of the Blessed Virgin into Heaven.

House of the Virgin Mary in Ephesus
In her visions, Sister Emmerich described with great detail the geography of Ephesus, its location to the sea and the many roads leading into the major port city in Asia Minor. She mentioned many caves in the mountain. The location was described as “some three and a half hours from Ephesus.” Today, the car ride from Ephesus to Meryem Ana Evi, up the paved road to Bulbul Dagh, is less than thirty minutes. But in the 1st Century, the trek up the mountain side would have taken several hours. In describing the mountain, Emmerich says that, “This hill slopes steeply towards Ephesus,” which Bulbul Dag does as it rises to the south of the ancient city of Ephesus.
In Blessed Emmerich’s visions, Mary was visited on occasion by “an Apostle or disciple on his travels,” and, of course, Emmerich saw one man going in and out of the house more often than others, and said, “I always took him to be [Saint] John.” She had a revelation that St. John did not remain permanently near the Blessed Virgin, but, “He came and went in the course of his travels. Emmerich said of the house St. John had built for the Blessed Virgin that it was unique among the village in that, “Mary’s Home was the only one built of stone . . . built of rectangular stones, rounded or pointed at the back.” She said that, “Between the Blessed Virgin’s dwelling and Ephesus runs a little stream which winds about in a very singular way.”
Sister Emmerich accurately described the view from the top of the mountain on which Mary’s Home was built. She said, “A little way behind it was the summit of the rocky hill from which one could see over the trees and hills to Ephesus and the sea with its many islands.” - The House of the Blessed Virgin, Mary (Selçuk, Turkey)
Further reading can be pursued with the following sources of information: