Market benchmarks in Tokyo, Hong Kong and Seoul declined. Shanghai swung between gains and losses.
Overnight, Wall Street's benchmark S&P 500 index broke a seven-day streak of record closes and fell after the Institute of Supply Management reported service industry activity grew in June at a slower rate than forecast.
The “disappointing drop” suggests the US economic recovery “is not immune” to global pockets of resurgence of the coronavirus, said Mizuho Bank in a report.
The Nikkei 225 in Tokyo sank 1 percent to 28,363.82 and the Hang Seng in Hong Kong lost 0.7 percent to 27,881.92.
The Shanghai Composite Index was up 0.2 percent at 3,535.34 at mid-morning after China's Cabinet announced it would impose stricter data security and other standards on Chinese companies that want to join foreign stock exchanges.
The announcement, at a time when Beijing is tightening control over technology industries, is a potential hurdle for Chinese entrepreneurs who have raised billions of dollars abroad. It comes after ride-hailing service Didi Global Inc was ordered to stop signing up new users and remove its app from online stores while it increases security for customer information.
The Kospi in Seoul retreated 0.6 percent to 3,287.11 while the S&P-ASX 200 in Sydney gained 0.7 percent to 7,316.00.
New Zealand, Singapore and Jakarta declined.
On Wall Street, the S&P lost 0.2 percent to 4,343.54 on Tuesday, led by losses for banks and energy companies. The index is up 15.6 percent for the year.
The Dow Jones Industrial Average fell 0.6 percent to 34,577.37. The Nasdaq Composite rose 0.2 percent to 14,663.64.
The ISM purchasing managers'' index fell to 60.1 from May's record 64.0 on a 100-point scale on which numbers above 50 show activity increasing. That was well below the 63.3 expected by forecasters surveyed by The Wall Street Journal.
Travel, hospitality and other services industries have enjoyed a boom as US restrictions on consumer activity ease.
That pushed up US prices, but the latest measure could support the Federal Reserve's position that the inflation spike is temporary. That could help to reassure investors the Fed and other central banks won't feel pressure to cool price rises by rolling back economic stimulus.
Also Tuesday, Didi shares dropped 19.6 percent in New York. That follows a 5 percent drop on Friday after Chinese regulators said they were investigating information security at Didi and two other ride technology companies.
Full Truck Alliance, the operator of two truck logistics platforms, lost 6.7 percent and Kanzhun Ltd, an online recruitment outfit, dropped 15.9 percent.
Amazon jumped 4.7 percent after the Pentagon said it is cancelling a cloud-computing contract with rival Microsoft that could eventually have been worth $10 billion and will instead pursue a deal with both Microsoft and Amazon. Microsoft shares were little changed.
In energy markets, benchmark US crude lost 14 cents to $73.23 per barrel in electronic trading on the New York Mercantile Exchange. The contract fell $1.79 to $73.37 on Tuesday. Brent crude, the basis for pricing international oils, retreated 30 cents to $74.23 per barrel in London. It plunged $2.63 the previous session to $74.53.
The dollar edged down to 110.60 yen from Tuesday''s 110.63. The euro declined to $1.1824 from $1.1826
