Does using Caitlyn's ability exhaust the champion, and does this prevent her from moving in/out of battlefields?

Ruling: Yes, Caitlyn's ability has the exhaust icon, so using it exhausts her. After using her ability, she cannot use a standard move action (which requires exhausting) unless she is readied again.

Sequence:

• Using Caitlyn's ability exhausts her
• An exhausted unit cannot perform actions that require exhaustion (like standard movement)
• If Caitlyn is readied after using her ability, she can exhaust to move again

Nuances:

• Only standard movement exhausts a unit; if a card effect moves a unit (like Charm or Yasuo's ability), it does not exhaust that unit
• Abilities without the exhaust symbol can be used even when the unit is exhausted
• Units can potentially use exhaust abilities more than once per turn if they can be readied multiple times