Per the Lodestone, a Moogle Treasure Trove event will be coming to Final Fantasy XIV on Thursday, May 30th and will end with the launch of the Shadowbringers expansion. This event, according to Square Enix, has no set schedule, so it may be reinstated at varying dates in the future.
In this event, certain duties will reward Irregular Tomestones of Philosophy, which can be exchanged with an Itinerant Moogle in Eorzea’s initial three city-states. Rewards include exclusive items like a Namazu Earring, but also harder-to-get items like lanner whistles and riding maps.
Duties that qualify for the Moogle Treasure Event will have a moogle icon next to them. Note that they must not be in-progress duties or with undersized parties.
You can check out the full list of rewards here.