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An encouraging platform was mostly squandered as the hosts' spin trio halted the Black Caps' progress in Kanpur, the tourists plummeting from 197-1 to be bowled out after tea for 296 in reply to India's 345.
At stumps, India reached 14-1, leading by 63 runs, and with any fourth-innings chase over 200 set to be devastatingly difficult, the Black Caps will need to replicate their brilliant day two performance when they resume on day four.
Day three started with the Black Caps on 129-0, and they reached 151-0 before Will Young (89) departed. Neither he nor Tom Latham (95) were able to advance to three figures, left to watch instead as the senior trio behind them undid some of the good work.
Axar Patel (5-62) was chiefly responsible for that as New Zealand suffered through a 26-over spell in which they lost 5-45, the left-armer taking the scalps of Ross Taylor, Henry Nicholls and Latham.
Ravi Ashwin and Ravindra Jadeja also chimed in with a wicket apiece as the Black Caps slumped to 249-6 at tea, and despite some defensive tail resistance, runs were hard to come by as their innings ended 49 runs short of India's total.
Young added only 14 to his overnight tally before edging behind off Ashwin's bowling - the increasing turn a harbinger of what was to follow - but that was enough to give him a new high score.
Latham, meanwhile, continued to show why he is regarded as one of the Black Caps' best against spin, looking comfortable and well on course for his 12th test century.
Given only three New Zealand batsmen had entered this series with past test experience in India, and given Kane Williamson and Taylor would both fall cheaply, the Black Caps owed Latham a debt of gratitude regardless of his eventual stumping.
He received a life when struck in front by Ashwin on 66, given not out before India inexplicably opted against reviewing, with replays showing the delivery would have cleaned up middle and off.
Latham had advanced to 82 by lunch but the meal wouldn't have been quite as palatable after he lost his partner just before the break.
In the second over with the new ball, Umesh Yadav got one to seam in sharply enough to break through Williamson's defences, sending the skipper on his way for 18.
That vital wicket gave India the edge for the first session, one they went about extending in the second.
Taylor had already survived one moment of peril while facing Patel, with substitute wicketkeeper KS Bharat unable to complete a tough stumping chance. But Bharat quickly redeemed himself by taking a fine catch when Taylor sought to defend, beaten by a bit of turn to fall for 11.
Nicholls' stay was brief as Patel picked up a second, the ball keeping low to sneak through the batsman's attempted slog sweep and strike the pad.
Latham was the next to fall, leaving his crease in a defensive attempt and being deceived by the turn, before Jadeja bowled debutant Rachin Ravindra for 13 to complete the collapse.
After tea, Tom Blundell was unlucky to see a delivery from Patel shoot along the ground and skittle his stumps, but with balls staying low all test, it made his decision to painstakingly bat 94 balls for 13 runs questionable at best.
Kyle Jamieson was similarly stoic on an admittedly difficult surface to score on, but after grafting 23 runs off 74 balls he either changed strategy or lost patience on the 75th, and an almighty slog was caught on the midwicket boundary.
New Zealand's innings wrapped up soon after that, but Jamieson was soon back in the action, clean bowling Shubman Gill with his first ball.
In just his ninth test, Jamieson became the fastest New Zealander to 50 test wickets, and he'll likely need to pick up a few more on day four to keep the Black Caps in with a chance of victory.