Question
What do you do with a mole after you trap it?

Answer
Assuming you live in a state where it is legal to trap moles (Massachusetts and Washington are two states that have banned the most effective mole traps thanks to their wonderful voters), the mole should be dead. If it is, then you can leave in the ground or dispose of it according to your town's waste disposal guidelines.
Be sure you wear gloves when handling all traps and wildlife as animals carry diseases contagious to humans.