Souls can be independent of an individual body; the animals themselves may contain souls; potentially even the souls/demons that are the agents of the devil.
Mat 8:28 And when he had come to the other side, to the country of the Gadarenes, there came out to him from the place of the dead, two who had evil spirits, so violent that no man was able to go that way.
Mat 8:29 And they gave a loud cry, saying, What have we to do with you, you Son of God? Have you come here to give us punishment before the time?
Mat 8:30 Now there was, some distance away, a great herd of pigs taking their food.
Mat 8:31 And the evil spirits made strong prayers to him, saying, If you send us out, let us go into the herd of pigs.
Mat 8:32 And he said to them, Go. And they came out, and went into the pigs; and the herd went rushing down a sharp slope into the sea and came to their end in the water. – Gospels According To Matthew
An unclean spirit, departing the body, perhaps on death, obstructed, I would imagine, from the Kingdom of God, has nowhere to return but back to the Earth. Suggesting reincarnation. Also suggesting that return trips accrue corruption.
Mat 12:43 But the unclean spirit, when he is gone out of a man, goes through dry places looking for rest, and getting it not.
Mat 12:44 Then he says, I will go back into my house from which I came out; and when he comes, he sees that there is no one in it, but that it has been made fair and clean.
Mat 12:45 Then he goes and takes with him seven other spirits worse than himself, and they go in and make it their living-place: and the last condition of that man is worse than the first. Even so will it be with this evil generation. – Gospels According To Matthew
Luk 11:24 The unclean spirit, when he has gone out of a man, goes through dry places, looking for rest; and when he does not get it, he says, I will go back to my house from which I came.
Luk 11:25 And when he comes, he sees that it has been made fair and clean.
Luk 11:26 Then he goes and gets seven other spirits more evil than himself, and they go in, and take their places there: and the last condition of that man is worse than the first. – Gospels According to Luke