When going away from a tamed mob, like a cat or parrot, once you travel a long distance away, they will not teleport to you, this happens while they are standing up
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I've also had the issue, on Android. In my case, it was a dog not teleporting to keep up with me in a boat.
This is usually expected behavior. Mobs (and other entities) become suspended when their distance from the nearest player exceeds your Simulation Distance setting. In the suspended state, they cannot do anything or change in any way. This is especially noticeable when you're in a boat, because pets will not teleport into water and there is often no solid ground near your boat that they can teleport to.
However, if this happens while you're still within simulation distance and there is solid ground near you, this might actually be a bug. Please comment if so, giving your Simulation Distance setting, the kinds of blocks you would expect them to teleport to, and their distance from your position. Also update us on which affected version you are using now. This ticket will automatically reopen when you reply.
this happened to me too. my dog did not teleport to me after crossing two islands.when i went back the dog was just in the water floating at the same spot. and i lost him again.