Their second time of 44.146 seconds was slower than their first, their cause not helped by Dawkins losing touch through the middle stages.
It left them 0.712sec outside a berth in the final, which was to be contested between the Netherlands and Great Britain. Russia and France were to race for bronze, with Germany fifth.
The women's sprint team of Natasha Hansen and Emma Cumming finished higher than the men, ranked fifth after missing out on a ride for bronze by 0.158sec.
They were seventh-fastest in the heats but improved in the first round, where China narrowly pipped them for fourth.
Germany and the Netherlands were to contest gold.
There was further New Zealand disappointment in the team pursuit, where hopes of another double medal haul were ended.
Eager to at least match their silver-bronze outcome at Hong Kong last year, the Kiwi men were shut out of the medals while the women scraped through to the semi-finals.
The men, who were runners-up a year ago, will ride off against Switzerland for fifth.
They fielded a near-identical lineup to Hong Kong, with Regan Gough, Dylan Kennett and Nick Kergozou joined by Campbell Stewart, who replaced Pieter Bulling.
However, they were pipped by 0.175sec by Germany for the last spot in the final four.
Their time of 3 minutes 57.622 seconds was less than 2sec behind top qualifiers Great Britain, with Italy, Denmark and Germany booking the other slots in a closely-fought qualifying session.
The Kiwi women also featured one change from the group who snared bronze last year, with Bryony Botha replacing Jaime Nielsen, alongside Racquel Sheath, Rushlee Buchanan and Kirstie James.
They clocked 4min 23.530sec to sneak into fourth but were nearly 5sec slower than top qualifiers the United States, who they will meet in the semi-finals.
