What happens to a moon when its planet is stolen?

We have known for a while that the Milky Way is full of homeless worlds. Rogue planets, free-floating planets—call them what you like—but they are planets formed in orderly systems around ordinary stars and then thrown out, either by a gravitational shove from a sibling planet or by a star that strayed too close. What has been much less clear is what happens to any moons they were carrying at the time. Eviction is a violent business, and you would expect a moon to be the first thing lost.