The study looked at the three main ranking systems - Quacquarelli Symonds (QS), Times Higher Education (THE) and the Academic Ranking of World Universities (ARWU).
All the top-placed New Zealand universities fell down the QS rankings, but the picture was more mixed in the other two systems.
The University of Auckland has remained relatively stable in both the ARWU and THE rankings. Otago, and more recently Canterbury, improved in the ARWU rankings but Massey and Victoria University of Wellington fell.
The report also looked at the performance of Australian universities, and found that all dropped in the QS rankings between 2007 and 2013.
"The rise of universities from Asian countries in the rankings is one factor in displacing the Australasian universities," it concluded.
University of Auckland vice-chancellor Stuart McCutcheon said although the performance of New Zealand universities was improving, international competitors were getting better faster.
That could affect international student numbers, which universities are trying to increase after a change in government effectively capped domestic numbers, and the funding that goes with them.
He said income per student had declined in real terms, and New Zealand universities brought in less income from government funding, fees, philanthropy and research.
Tertiary Education Minister Steven Joyce said the Government had significantly increased funding to the tertiary sector, which was high in the OECD as a proportion of GDP.
Some universities had lagged in international enrolments, he said. The University of Auckland brought in international income of about $80 million in 2012, compared with $267 million at the similarly sized University of Queensland.
"I'm never going to be able to close that gap for them, they have to get out and close that gap for themselves ... They are starting to do that, but they need to do it faster."
Struggling to compete
• Long-term analysis of international university rankings finds all New Zealand institutions fell in one survey, with results more mixed in the other two main rankings.
• Australian universities also fell in some surveys, with the report concluding that the rise of Asian rivals is one factor.
• Controversy over how rankings are calculated, but "they are important, ultimately, because people think they are important'', says ministry.