Investors were also keeping track of developments in Donald Trump's trade war and signs of movement on possible talks between the US president and his Chinese counterpart Xi Jinping.
Wall Street provided an uninspiring lead as a report by payroll firm ADP showed private-sector jobs rose by 37,000 last month, a sharp slowdown from April's 60,000 and less than a third of what was forecast in a Bloomberg survey.
Another survey showed activity in the services sector contracted in May for the first time since June last year.
The readings stoked concerns that the world's number one economy was stuttering, with the Fed's closely watched "Beige Book" study noting that "economic activity has declined slightly".
It flagged household and business caution caused by slower hiring and heightened uncertainty surrounding Trump's policies.
However, the readings ramped up bets on a Fed cut, with markets pricing in two by the end of the year, with the first in September.
Eyes are now on the non-farm payrolls release on Friday, which the central bank uses to help shape monetary policy.
Still, there is some concern that the US president's tariff blitz will ramp up inflation, which could put pressure on the Fed to keep borrowing costs elevated.
Most of Asia rose, with Hong Kong, Shanghai, Singapore, Taipei, Mumbai, Bangkok and Wellington up with London, Paris and Frankfurt.
Seoul rallied more than one percent on continued excitement after the election of Lee Jae-myung as South Korea's new president. The vote ended a six-month power vacuum sparked by the impeachment of predecessor Yoon Suk Yeol for a calamitous martial law attempt.
The won rose around 0.3 percent, building on a recent run-up in the currency against the dollar.
Jakarta advanced as Indonesia's government began rolling out a $1.5 billion stimulus package after Southeast Asia's biggest economy saw its slowest growth in more than three years in the first quarter.
Tokyo fell following another weak sale of long-term Japanese government bonds, which added to recent concerns about the global debt market.
The soft demand also stoked speculation that the government could scale back its auctions of long-term debt in a bid to boost demand.
Investors are awaiting news of talks between Trump and Xi, with the White House saying they could take place this week.
But while tariffs remain a millstone around investors' necks, IG's chief market analyst Chris Beauchamp said traders seemed less concerned than they were after the US president's April 2 "Liberation Day" fireworks.
"With markets still rising, the overall view appears to still be that the US is no longer serious about imposing tariffs at the levels seen in April," he wrote in a commentary.
"President Trump appears fixated on a call with China's president that might help to move the situation forward, but Beijing remains wary of committing itself to any deal.
"This does leave markets open to another sudden shock, which might replicate some of the volatility seen in April. But that manic period appears to have dissuaded the administration from further major tariff announcements."
- Key figures at around 0810 GMT -
Tokyo - Nikkei 225: DOWN 0.5 percent at 37,554.49 (close)
Hong Kong - Hang Seng Index: UP 1.1 percent at 23,906.97 (close)
Shanghai - Composite: UP 0.2 percent at 3,384.10 (close)
London - FTSE 100: UP 0.1 percent at 8,807.05
Euro/dollar: DOWN at $1.1415 from $1.1417 on Wednesday
Pound/dollar: UP at $1.3566 from $1.3548
Dollar/yen: UP at 143.29 yen from 142.86 yen
Euro/pound: DOWN at 84.15 pence from 84.26 pence
West Texas Intermediate: UP 0.3 percent at $63.03 per barrel
Brent North Sea Crude: UP 0.4 percent at $65.10 per barrel
New York - Dow: DOWN 0.2 percent at 42,427.74 (close)
dan/dhc
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