CARLOW, Ireland—The people of Ireland go to the polls Friday to deliver what's expected to be a knock-out blow to the governing party. But many are choosing to vote in a traditional Irish fashion: with their feet. Tens of thousands are joining in a new wave of emigration, turning their backs on a country mired in economic malaise.

Martin Lynch's family is one of many that are being scattered to the four winds. In the past year, one son has moved to Germany, another to England. His daughter is planning her departure for London,...