When Buneary evolves into Lopbunny, Run Away will become Cute Charm. Buneary can have 2 abilities naturally (excluding its Dream World ability Limer), which are Run Away and Klutz. Lopbunny can have Cute Charm or Klutz. Therefore a Buneary with Klutz will have Klutz as a Lopbunny, but a Buneary with Run Away will gain Cute Charm.
The only way a Pokémon can change its Ability is upon evolution. The "slot" storage system means that if an unevolved Pokémon has the "first" ability for its species, it will have the "first" ability of its evolved species upon reaching this stage, even if this Ability differs in the evolved species. In other words, a Poochyena with Run Away would evolve into a Mightyena with Intimidate, while a Poochyena with Quick Feet would evolve into a Mightyena with Quick Feet. If the unevolved Pokémon has only one Ability, but its evolution may have one of two Abilities, then it has a 50% chance of acquiring either Ability (dependent on its personality value).
So with Aipom, Run Away will become Technician, and Pickup will remain as Pickup when it evolves into Ambipom.
Bulbapedia on Abilities