Kent recovered well after early losses but were bowled out for 162 and 293 at the end of day three.
Chris Read was in solid form behind the stumps catching captain Rob Key, Joe Denly and Matthew Walker.
Will Jefferson caught Geraint Jones at second slip after a parry, his second catch in quick succession having seen off Darren Stevens minutes earlier.
Kent were 104-5 at lunch having posted a first innings total of 162 all out.
Azhar Mahmood made his century before play was stopped at 7.15pm.