Relocating elite moving between Magdalen and Christ Church? (15)

I believe the answer is:

intercollegiate

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'magdalen and christ church?' is the definition.
I can't judge whether this defines the answer.

'relocating elite moving' is the wordplay.
'moving' is an anagram indicator (letters are moved around).
'relocating'+'elite'='relocatingelite'
'relocatingelite' anagrammed gives 'INTERCOLLEGIATE'.

'between' is the link.

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I've seen this clue in The Times.

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