From Bulbapedia, the community-driven Pokémon encyclopedia.
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Chain breeding is the process of breeding a move to a Pokémon that can only learn it as an Egg Move, then in turn using that Pokémon to pass down the move to another which can learn the move as an Egg Move, but cannot breed with the original Pokémon.
Many moves must be chain bred, with these moves being more prevalent in earlier generations, as more Pokémon with larger learnsets and different Egg Group combinations continue to be introduced. What follows is a list of the steps needed to get an Eevee with Wish in Generation V:
It can also be used to pass on entire movesets, even if individual moves are able to be passed down without an intermediary. For example, taking the child Eevee with Wish and leveling it up so that it will learn Baton Pass at level 36, it can pass down both moves to a child Spinda, which cannot learn either of them normally.
Many breeding chains, however, are far longer than the Wish chain, and breeding chains can get longer still if a specific set of four moves is wanted on a specific Pokémon. Most chains that begin in the Field Egg Group are made easier by Smeargle's ability to learn any move through Sketch, and some, especially in early generations, explicitly require that a Smeargle Sketch a move that is otherwise exclusive to an all-female or genderless species, such as Heal Bell in Generation II (exclusive to Miltank and Celebi).