Resuming 506 runs behind on 69-1, Chris Nash moved on to 55 but was then removed lbw by Kagiso Rabado (3-81).
Luke Wright (60) and Taylor led a revival but Sussex collapsed from 219-3 to 273-7 after Kent took the new ball.
Danny Briggs, Steve Magoffin and Stu Whittingham were unable to stick with Taylor and Kent look set to enforce the follow-on, with Sussex 242 runs behind.
Sussex were bowled out for 333 on day three by Kent, despite Ross Taylor's 142 not out at Tunbridge Wells.