(The rabbit in the picture is tamed, don't ask)
It seems when you're far away from a rabbit, it won't teleport like wolves and ocelots do. Also, when you're far away and one block above it, it will stop following you altogether.
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I don't know why MC-60123 was closed as a Duplicate. It was opened first and has more comments, Votes, and watchers.
Reposting my comment.
Farm animals (cow, pig, sheep) don't follow unless you have a lead or the food they like. Rabbits should adhere to this convention.
Dogs and cats are "useful" when they follow you. (Increased attack & HP, frighten creepers)
There is no reason for rabbits to follow you around.
They should behave like horses, they require taming, but have no owner.
I believe this is a duplicate of MC-55661.
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