Wayne Madsen hit 93, but five wickets in the day for Adam Riley has put Kent in control after Day Three at The 3aaa County Ground.
Madsen and Alex Hughes (66) shared a 130-run partnership for the third wicket, but after both fell just before lunch Derbyshire lost regular wickets to be dismissed for 400 – a first innings deficit of 161 runs.
Kent enforced the follow-on and Riley, who took three in the first innings earlier in the day, claimed two more wickets to leave Derbyshire 83-2 at stumps, still trailing by 78 runs.
The day started well for the hosts as Madsen and Hughes took the overnight score from 210-2 to 288-2 after the first 90 minutes of play.
Hughes reached his half-century off 108 balls, with six fours, but his Madsen fell shortly after just seven runs short of a third century of the summer. The veteran batsman was lbw to Riley for 93 off 157 balls, with nine fours.
Riley claimed his second not long after as Hughes was caught at short mid-wicket for 66 off 161 balls, with Derbyshire 308-4 at lunch.
Kent then dominated the afternoon session taking six wickets as the hosts slipped to 400 all out.
Sean Ervine hit 26, but no other batsmen passed 20, with the last wicket stand of 30 being the highest partnership in the session. Still trailing by 161 runs, Derbyshire were asked to bat again.
Billy Godleman and Tom Lace for the second time in the match put on 50 for the first wicket, but Godleman was dismissed soon after as he was bowled by Riley for 26.
The spinner was at it again four overs later when Madsen (5) hit a powerful drive, but it was straight back to the bowler who took a good low catch. Derbyshire closed on 83-2, with Lace 28 not out and Hughes unbeaten on 16.
Four-day cricket is back! Derbyshire host three Specsavers County Championship matches in August and September. Buy your tickets in advance and save.