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There is a document called the protoevangelium (prequel to the gospel) of St James, the earliest copy of which is said to be third century.
This describes it in more detail.
Chapter 22 verse 3 says
But Elizabeth when she heard that they sought for John, took him and went up into the hill-country and looked about her where she should hide him: and there was no hiding-place. And Elizabeth groaned and said with a loud voice: 0 mountain of God, receive thou a mother with a child. For Elizabeth was not able to go up. And immediately the mountain clave asunder and took her in. And there was a light shining alway for them: for an angel of the Lord was with them, keeping watch over them.
It goes on to say Zacharias, John's father, was subsequently killed for refusing to reveal John's whereabouts.
This is not regarded as canonical.
Ein Karem is about five miles from Bethlehem though, so would not necessarily have been included in the death zone.
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Actually itβs in the Bible and often overlooked, I recommend starting at Verse 1 to get the full context of one of the most beautiful Chapters written imo; specifically the last half of this verse
Luke 1: 80 And the child grew, and waxed strong in spirit, and was in the deserts till the day of his shewing unto Israel.
Clearly the hand of God was upon John to fulfill His Will, and the Holy Spirit guiding and protecting him (of which both Zacharias his father and Elisabeth his mother were also filled with) just as when Jesus being annointed by God with the Holy Spirit without measure after being baptised by John then also went into the wilderness/deserts carrying only his faith and trust in the Father with him to be tempted of the devil.
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Catholic.com answers this quite simply from the reasoning point of different geographical location.
Scripture does not say, but Herod apparently limited the murders to the region (Matt. 2:16) where he knew the Messiah was to be born (Matt. 2:1-6). So far as we know, St. John the Baptist and his parents did not live in Bethlehem or its immediate vicinityβotherwise Mary and Joseph could have stayed with them while in the area for the enrollment (Luke 2:4-7)βbut in the hill country, in an unnamed city of Judah (Luke 1:39-40), which may have been distant enough from the attack directed by Herod to protect the infant John.