Elite institutions that beast can cross? (8)
I believe the answer is:
oxbridge
'elite institutions' is the definition.
'beast can cross?' is the wordplay.
'beast can' becomes 'ox' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'cross?' becomes 'bridge' (bridging is a kind of crossing).
'ox'+'bridge'='OXBRIDGE'
'that' is the link.
(Other definitions for oxbridge that I've seen before include "Model of academic elitism" , "Universities united" , "Home of the Boat Race crews" , "linking two top-quality bodies" , "The two most famous British universities regarded like 32 Across" .)