Animals are legally speaking, property, so they can, and frequently are, gifted in a person's last will and testament.
Where an animal is given as part of a will, the executor has a duty to transfer the animal (in your case a dog) as soon as possible and then he or she will report the transfer to the court in the probate process. That is, the dog gets transferred to the heir now and is documented for having been transferred later on.
We always think of the probate process as cumbersome and time consuming, and it can sometimes be. Probate is however, really useful in getting real estate or motor vehicles retitled since the court will pave the way for the heir to receive a new deed or title to the property after the probating is done. However, since animals have no titles or deeds, they may be transferred under the will provision immediately and there is usually no need wait for probate to close up and become final before handing the dog over. The only exception I can think of is where the transfer of the animal is disputed, such as in someone challenging the will.
If you want your dog to go to a particular person, then your will should state that. The best way to handle the transfer of pets after their owner's death is through a trust. You should speak with the attorney who drafted your will about setting up a trust for your pets. This is not something you should attempt to handle yourself as the will and trust rules are very specific and if you make a mistake, the entire document may be void.