Increases being, in various cases, overdue (9)
I believe the answer is:
escalates
'increases' is the definition.
(escalating is a kind of increasing)
'in various cases overdue' is the wordplay.
'in' is an insertion indicator.
'various' indicates an anagram.
'overdue' becomes 'late' ('late' can be similar in meaning to 'overdue').
'cases' with letters rearranged gives 'escas'.
'escas' going around 'late' is 'ESCALATES'.
'being' is the link.
(Other definitions for escalates that I've seen before include "Steps up" , "Increases in intensity" .)