Eight different bowlers struck as Kent dismissed Leeds-Bradford MCCU before building a 213-run lead.
Stand-in skipper Will Gidman rotated the attack after an overnight declaration did not pay immediate dividends.
The students’ opening pair put on 58 runs in 23.2 overs to see off the new ball before Mitch Claydon removed Steve Bullen to break the deadlock, caught behind by Adam Rouse.
Gidman brought himself and Adam Ball into the attack with an instant impact.
Ball trapped new man Martin Andersson LBW and Gidman uprooted McDermott’s middle stump in successive overs.
Stevens dislodged Oliie Graham’s off stump before Ivan Thomas found the outside edge of opener Ben Shoare for 47.
The right-hander had scored a century last summer against Kent’s second XI playing for MCC Young Cricketers and struck seven boundaries in a patient knock.
Claydon returned to remove Craig Wallace, caught smartly at slip by Ball and Jack Potticary fell in similar fashion, caught Tredwell off Haggett.
Tredwell bowled Will Cook after his 56-ball resistance ended with just three runs to his name as the students struggled to avoid the follow-on.
Joe Denly eventually grabbed the final wicket after a 38-run last wicket stand as injured skipper Moin Ashraf scored 24 with the aid of a runner.
Kent opted not to enforce the follow-on and stretched their lead to 213 by stumps after Sean Dickson was bowled for 4 after a brief delay for bad light.
Ball (23*) and Gidman (17*) are set to resume on the third and final day on Thursday at 11am. Entry and parking at The Spitfire Ground is free of charge.