After the touring batsmen had a first taste of competitive action on Saturday, reaching 289 for five declared, their bowlers enjoyed a productive opening session on day two to leave the hosts 129 for six at lunch.
With Derbyshire resuming on 24 for one, Neil Wagner removed Billy Godleman (14) and Wayne Madsen (12), before Bracewell had Dan Redfern and Ross Whiteley caught in the slips from successive deliveries.
Wes Durston made a solid 46 to give the innings some backbone but was stumped just before lunch after carelessly charging Bruce Martin. Captain Wayne Madsen hit the first boundary of the day from the eighth ball but it did not take Wagner long to strike for New Zealand.
Godleman added five to his overnight score of nine before he feathered a good length delivery to wicketkeeper BJ Watling, who took a low catch.
Durston was next in and Wagner almost struck him on the thumb by way of greeting. Bracewell was operating from the other end, but a set of four byes over Watling's head was the most noteworthy aspect of his first spell.
Kane Williamson put down a sharp chance off Durston, who was on six at the time, but if Wagner felt aggrieved by the let off he had his third wicket of the innings four balls later. Madsen was the victim, nicking to Watling on the back foot.
Durston looked more assured at the other end, scoring four boundaries off Mark Gillespie to keep the board ticking.
Bracewell replaced Gillespie and immediately picked up two in two balls. The seamer, who missed the Test series against England earlier this year after cutting his foot following a party, produced two fine balls to see off Redfern (10) and Whiteley (nought). Both men were held in the slips as the hosts slipped to 89 for five.
Durston shepherded his side past three figures but fell before the interval when attacking Martin over-enthusiastically.