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...