Tamed animals that are standing will sit down when their owner changes dimensions and is no longer in the same dimension as them in version 1.21.4 and earlier.
In 1.21.5, however, the animals no longer sit, which means they are free to move as if they no longer belong to the owner.
This can be observed either by having a second player watch the owner's animals when they change dimension, or by using the sit/stand state detector (step-by-step shown in the attached video): https://www.youtube.com/watch?v=zakVncVTpgg
One small detail we can add in expected behavior: when the owner gets back in the same dimension as the dog, the dog should get back up on its own (as we can see in the attached video in the 1.21.4 part).