Kuldeep was the pick of the bowlers, finishing with 5-99.
Veteran Shaun Marsh is under pressure to retain his place in the team after scoring only 183 runs at an average of 26 in the four Tests.
Peter Handscomb was 28 not out and Pat Cummins was on 25, with Australia still needing another 287 runs to avoid the prospect of the follow-on after India posted 622-7 declared in their first innings.
Tim Paine's Australia touched a new low on Day four of the fourth Test against India when they were asked to follow-on by Virat Kohli after they got bundled out for 300 and trailed the tourists by 322 runs in Sydney.
It came after India also amassed a 292-run first-innings lead in Melbourne last week, only on that occasion they opted to bat again themselves in the 137-run flogging.
The recalled Peter Handscomb began with objective, confidently stroking two boundaries to move to 37 before swiping at a Bumrah delivery and dragging it onto his stumps.
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Openers Usman Khawaja (4) and Marcus Harris (2) safely negotiated four overs until bad light and light drizzle forced an early end to an already heavily rain-disrupted day just before the rescheduled tea break.
With 98 overs to be bowled in tomorrow's day play and if weather holds up, the Indian bowlers would be looking to finish the series on a high and help India create history by winning the series 3-1.
Nathan Lyon struggled in the few balls he faced against Kuldeep Yadav and fell for a duck, before Mitchell Starc (29) and Josh Hazlewood (21) offered some cavalier batting at the death to boost Australia's total. Yadav ended with 5-99.
Yadav then trapped Nathan Lyon (0) lbw as Australia lost three wickets for 22 runs in the space of 44 balls.
Kuldeep claimed three wickets and Jadeja two as Australia collapsed from 122-1 at lunch to 198-6 shortly after tea in a pattern all too familiar for an Australian batting order yet to score a century this series.