The health of the global economy comes down to a race between money flooding out of emerging markets amid the coronavirus pandemic and the efforts of the International Monetary Fund and World Bank to pump money back in.

The two Washington-based finance institutions find themselves facing the greatest challenge since they were established as the heart of the international monetary system at the 1944 Bretton Woods Conference. The IMF’s seminal mission was to safeguard global financial stability to prevent a repeat of the Great...