LIMA, Peru—Talk of the Federal Reserve’s first rate increase in almost a decade tends to send many investors into a frenzy. For the world’s central bankers, it is increasingly likely to elicit sighs of resignation.

Fed fatigue has enveloped emerging-market officials facing repeated bouts of volatility in their currencies and capital flows alongside mounting worries about debt. Some policy makers, gripped by the uncertainty, delivered a message to their American counterparts as officials gathered in the Peruvian capital for the International Monetary Fund’s annual meeting: Please stop dithering.

“Delaying the increase would not solve the situation,” said Sukhdave Singh, deputy governor of Bank Negara Malaysia. “If it is a case that the emerging markets have taken on too much debt, there will be a day of reckoning. Delaying an interest-rate hike does not necessarily address that issue.”

Once the Fed moves, investors will move money back into the U.S., depriving emerging markets of capital, which will weaken their currencies and send inflation higher.

Fed fears have consumed emerging economies for the past two years after an unprecedented stretch of U.S. monetary stimulus. But many officials at the IMF meeting, which ended Sunday, said they would prefer certainty now over the prolonged agony of waiting.

“This year, compared to a year ago, many emerging-market central bank governors and some others were keener that the Fed just get on with it, not because they were keen to see interest rates rise, but because they wanted to reduce uncertainty,” said Tharman Shanmugaratnam, Singapore’s deputy prime minister and former head of the IMF’s governing committee.

Emerging-market officials aren’t the only ones looking for the Fed to get on with it.

Jens Weidmann, president of Germany’s central bank, said the prospect of emerging markets getting hurt by a Fed-induced capital outflow is “no reason” to delay a rate increase that is justified by data. A U.S. rate increase “would be a reaction to a better economy and that would ultimately be good news for the world economy,” he added.

Fed officials, after fielding years of pleas to delay a rate increase, say they are proceeding cautiously and resting their decision on the strength of the U.S. economy.

“It has been clear from conversations at this conference that many officials of emerging-market and other countries feel sufficiently forewarned and prepared for them to want us ‘to just do it,’ ” Fed Vice Chairman Stanley Fischer said at a conference in Lima on Sunday. “However, we have to remain cognizant of the risks ahead.”

Mr. Fischer said officials don’t expect international developments to hurt the U.S. economy enough to change the central bank’s policy plans.

Many emerging-market central bankers say they have done everything they can to prepare for a U.S. rate increase. They let their currencies float, indicating a willingness to absorb higher inflation in exchange for cheaper exports. They accumulated foreign currency reserves to absorb the shock of fleeing capital. And many set inflation targets to reassure investors.

But those preparations won’t protect them from currency swings as money sloshes around the world at the merest hint of Fed action, driving up the value of the dollar and inflation in emerging markets. For those central bankers targeting an inflation rate, the swings can force difficult choices, such as raising their own interest rates despite slowing economies.

Not everyone is itching for Fed action.

Backers of further delay from the U.S. central bank have an important ally in the IMF, which says the world economy is still too weak to withstand a rate increase this year. Fed tightening could spur a wave of corporate defaults as companies borrowing money in dollars face higher debt costs, the IMF says.

Many economists were confident the Fed would move in September based on the strength of the U.S. economy. But the devaluation of the Chinese yuan and the turmoil in the Chinese stock market raised the specter of a deeper-than-anticipated slowdown. That stayed the Fed’s hand, though officials were quick to say they still intended to raise rates this year.

Since then, weak economic data in the U.S. and slower expected growth world-wide have shaken up expectations and paralyzed emerging markets.

One mitigating factor for emerging markets is that U.S. officials have been clear about their intentions to raise rates. That has given investors time to move their money and central bankers in the developing world to prepare. But not even the clearest communication can fully protect emerging markets.

Nobody can say they were caught by surprise,” said Ilan Goldfajn, chief economist at Itaú Unibanco. “That said, markets are crazy.”

Anybody looking for certainty once the Fed moves is bound to be disappointed, said Lesetja Kganyago, head of the South African central bank. The first Fed rate increase will only lead to uncertainty about subsequent moves, he said.

“It’s the uncertainty that seems to be a permanent feature now,” he said. “Volatility becomes the order of the day.”

Officials say they are watching every word from central bankers in the world’s largest economy. They face at least a few more months of waiting. “Everyone talked about September, then they were talking about December,” said the Paraguay central bank chief, Carlos Fernández Valdovinos. “Right now, here, after this meeting, everyone is talking about January.”

How do you prepare for such a hazy outlook? “I think the word for me is be smart, be wise, know your limits,” he said.

—Ian Talley, Paulo Trevisani and Todd Buell contributed to this article.