Shiv Thakor claimed career-best figures of 5-63 but Kent’s Sean Dickson scored an unbeaten double-century to help leave the match in the balance going into the final day.
Thakor became only the 11th Derbyshire player to score a hundred and take five wickets in the same innings as Kent were bowled out for 412 with Ben Cotton (3-66) picking up three wickets.
After securing a first-innings lead of 80 runs, Derbyshire stumbled towards the end of the day to reach stumps on 9-3 with a lead of 89.
Kent began Day Three on 79-1 with Dickson resuming on 31 not out and Sam Northeast replacing Joe Denly who was unavailable to continue his innings due to the birth of his child. He retired not out on 34.
Thakor picked up the first two wickets of the day when Northeast nicked him through to Poynton down the leg-side on 12 and Alex Blake edged him to Chesney Hughes at slip for six.
They were to be the only wickets of the morning as Kent reached lunch on 183-3 with Dickson playing well and unbeaten on 75 alongside Adam Ball on 35 not out.
Ball fell shortly after the break for 40 when Poynton took his third catch off the innings to give Cotton his second wicket. Cotton also took the next wicket when he nipped one back at Adam Rouse and castled him for 25.
Shortly after that, Dickson got through to his first century for Kent, which came from 243 balls, as he ensured Kent picked up three batting bonus points.
Thakor then produced a superb spell to take two wickets in as many balls, both lbw, to remove Calum Haggett for 15 and James Tredwell without scoring.
After tea, Dickson, along with Matt Coles, frustrated Derbyshire and ensured their side could not be asked to follow-on, by creating a 97-run partnership for the eighth wicket.
Coles made 41 before being bowled by Tom Taylor as the visitors edged past 400 with Dickson moving through to his double-century from 353 balls with 23 fours and two sixes.
The final wicket to fall gave Thakor his first five-wicket haul in first-class cricket. He bowled Mitchell Claydon without scoring as Kent were 412 all out, to give Derbyshire a first-innings lead of 80.
With six overs remaining in the day, Derbyshire began their second innings with Chesney Hughes and Billy Godleman both of whom were not out in the middle after three overs.
Godleman was forced off through an injury sustained from the second ball of the innings whilst Hughes was out lbw to Tredwell for a single.
Taylor had come in as nightwatchman to replace Godleman but was given out lbw to Coles without scoring and then Hamish Rutherford was caught behind off the same bowler also without troubling the scorers.
Derbyshire ended the day on 9-3, leading by 89 runs, with Wayne Madsen and Neil Broom not out on five and three respectively going into the final day.
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