UBS, Citi, Credit Suisse, HSBC, Julius Baer and Deutsche Bank have relied on Asia to power private banking revenues.

But investors there are shying away from riskier assets and new investments due to volatile markets. Asian stock markets have fallen as much as a fifth over the U.S.-Chinese trade row, touching off fears of an economic downturn.

"The recent emerging markets crisis and the trade war have definitely led clients to adopt a little bit more risk-off mode," said Kwong Kin-Mun, managing director, head of wealth management for Southeast Asia at Deutsche Bank.

After a strong first quarter and a steady second quarter for the industry, he said the third quarter of was slower than normal. He said the last three months was usually quieter too.

"There will still be opportunities for new relationships to be opened. In China, you get millionaires every day," he said. "That's why you see most of the banks are focussing a lot on China because that's where really the wealth is growing."

UBS CEO Sergio Ermotti said last week the world's biggest wealth manager faced declining revenue from transactions in the third quarter, saying clients were "much, much more insecure". But he said the rest of the business was holding up.

Falling transactions are not the only challenge. Lending income suffers when clients pare positions and fees recede as the value of assets dip.

Credit Suisse said it was also seeing a slowdown. "What you see lately is less activity, less transaction, definitely lower volume," a spokesman said, adding it was most noticeable in the Asia-Pacific region.

A senior official from a leading European private bank who asked not to be identified said Asian clients were de-leveraging due to the trade row and because of rising U.S. interest rates and the impact on emerging markets.

"We do see clients are becoming a bit more defensive in terms of their asset allocations and their short-term cash holding has been increasing," he said, although he said some investors were being attracted by the lower prices.

He said customers were putting less focus on equity positions and some had trimmed high-yield bond positions.

"In exchange, they are going for safer assets -- on the equities side more defensive stocks like the utilities and in the fixed-income space some of the shorter dated bonds or some hybrid bonds or some floating rate instruments," he said.

(Additonal reporting by Oliver Hirt and Brenna Hughes-Neghaiwi in Zurich; Editing by Edmund Blair)

By Sumeet Chatterjee and Angelika Gruber