Brown starts to take over after Blair's stint ends (5)
I believe the answer is:
toast
'brown' is the definition.
(I've seen this before)
'starts to take over after blair's stint ends' is the wordplay.
'starts to' says to take the initial letters.
'ends' suggests the final letters.
The first letters of 'take over after' is 'toa'.
The final letters of 'blairs stint' are 'st'.
'toa'+'st'='TOAST'
(Other definitions for toast that I've seen before include "Breakfast slice" , "A bit of a bloomer, possibly" , "Tribute (plus a clinking of glasses?)" , "Raise a glass to (someone)" , "Drink to someone" .)