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In 1492, the Ottoman Empire's Bayezid II sneered at Spain's Ferdinand of Aragon: "you call him a wise ruler, he who has impoverished his own country and enriched mine," with the expulsion of Jews and Moors. Bayezid sent warships to Spain to pick them up and take them to his lands, forcing his governors to accept them on pain of death. They contributed heavily to the finance and commerce of his lands.
Things had not changed much a century later, in 1609. The Ottoman Empire was "between" wars with the Hapsburgs. During the truces, their best "unofficial" weapon was the pirates of the Barbary Coast (modern Morocco and Algeria) that harassed commerce, particularly Spanish commerce, going through the Strait of Gibraltar.
Many of these pirates were European Christians called "renegades" that converted to Islam for the privilege of living off the Spaniards. Expelled Moriscos (not all of them were) fit right in, if not as pirates themselves, traders and artisans that supported them.