Can you play an exhausted unit to your base, move Miss Fortune to ready that unit, then move the newly readied unit to a battlefield as part of the same challenge/contest?

Ruling: No, you cannot do this in the same challenge/contest. Miss Fortune's move ability goes on the chain, and the unit only becomes ready once that ability resolves, which is after the showdown has already started.

Sequence:

• Play the exhausted unit to your base
• Move Miss Fortune to another enemy battlefield (her ready ability goes on the chain)
• Showdown begins
• Miss Fortune's ability resolves and readies the unit (too late to move it to the battlefield)