The Gospel of Matthew records that Eashoa' Ha Msheekha went to Eygpt with his Parents Joseph and Mary to flee the wrath of Herod.
Eashoa could have visted Ethiopia. There are so many accounts of Christ in the various Gospels, including the "non-canonical" Ones like the Gospel of Thomas. I think it is very likely that Eashoa visited Ethiopia.
Gospel of Matthew 12:12-23
 And being warned of God in a dream that they should not return to Herod, they departed into their own country another way.
 And when they were departed, behold, the angel of the Lord appeareth to Joseph in a dream, saying, Arise, and take the young child and his mother, and flee into Egypt, and be thou there until I bring thee word: for Herod will seek the young child to destroy him.
 When he arose, he took the young child and his mother by night, and departed into Egypt:
 And was there until the death of Herod: that it might be fulfilled which was spoken of the Lord by the prophet, saying, Out of Egypt have I called my son.
 Then Herod, when he saw that he was mocked of the wise men, was exceeding wroth, and sent forth, and slew all the children that were in Bethlehem, and in all the coasts thereof, from two years old and under, according to the time which he had diligently inquired of the wise men.
 Then was fulfilled that which was spoken by Jeremy the prophet, saying,
 In Rama was there a voice heard, lamentation, and weeping, and great mourning, Rachel weeping for her children, and would not be comforted, because they are not.
 But when Herod was dead, behold, an angel of the Lord appeareth in a dream to Joseph in Egypt,
 Saying, Arise, and take the young child and his mother, and go into the land of Israel: for they are dead which sought the young child's life.
 And he arose, and took the young child and his mother, and came into the land of Israel.
 But when he heard that Archelaus did reign in Judaea in the room of his father Herod, he was afraid to go thither: notwithstanding, being warned of God in a dream, he turned aside into the parts of Galilee:
 And he came and dwelt in a city called Nazareth: that it might be fulfilled which was spoken by the prophets, He shall be called a Nazarene.
Gospel of John 21:25
 And there are also many other things which Jesus did, the which, if they should be written every one, I suppose that even the world itself could not contain the books that should be written. Amen.
That verse of the Gospel of John implies that there are many events of Eashoa's Life that were never recorded, as it says "even the world itself could not contain the books that should be written".