[What Are Expedition Limimin?]
The Limimin floating around above your player in the Lobby can be sent out on Expeditions.
From these Expeditions you can earn three different kinds of rewards.

Expedition A: Gold is easier to obtain

Expedition B: Augment materials are easier to obtain

Expedition C: Reforge materials are easier to obtain

[Sending Limimin Out on an Expedition]
If your Limimin is not currently out on an Expedition already, tap on the Limimin floating around above your player in the Lobby to send him off. He will return from its Expedition in three hours.

[Viewing Expedition Results]
When the Limimin is out on an Expedition, there will be a countdown timer in the top-right hand side of the screen.
When the timer is up, the Limimin will return to the lobby and begin floating around your head again. Tap on him to bring up the list of rewards that he has brought back.
You can spend one Gem to receive three times the rewards.