Pupils having arrived initially at school in carriage (7)
I believe the answer is:
phaeton
'carriage' is the definition.
(thesaurus)
'pupils having arrived initially at school' is the wordplay.
'initially' indicates taking the first letters.
'at' is a charade indicator (letters next to each other).
'school' becomes 'eton' (famous UK public school**).
The first letters of 'pupils having arrived' is 'pha'.
'pha'+'eton'='PHAETON'
'in' is the link.
(Other definitions for phaeton that I've seen before include "transport" , "4-wheeled carriage" , "Coach" .)