“In calling the animal ‘Soulless,’ it is not depriving the beast, from the humblest to the highest species, of a ‘soul,’ but only of a conscious surviving Ego-soul, i.e., that principle which survives after a man, and reincarnates in a like man. The animal has an astral body, that survives the physical form for a short period; but its (animal) Monad does not re-incarnate in the same, but in a higher species, and has no ‘Devachan’ of course. It has the seeds of all the human principles in itself, but they are latent.” (“The Secret Doctrine,” vol. 2, p. 196 fn, Blavatsky)
What made me wonder about animal reincarnation was living in the country and feeding some semi-feral cats and noticing the similarity of different generation’s cat’s personalities (and they have definite personalities!) (They breed fast and die young, so a new generation is mostly every few years.) Recently I saw a transfer of behavior when it couldn’t possibly have been learned. I had cut a small hole in a door to the garage to let the cats get out of the weather, but boarded it up a year or so later after being tired of mucho cat guano in the garage. They usually headed for this hole like lightning whenever I was around to get away from me. (Cats like to play games and freak themselves out. Sometimes they could care less if I was around and lounge around and soak up the sun. Other times I make an appearance and they fly up in the air and take off pursued by demons.)
Awhile ago I decided to try to pick up a 1-pound about 2 month old kitten sitting in front of the garage door with the boarded-up former hole. He was a 1-pound terror and sliced and diced my hand pretty good when I reached for him, but the surprising thing was that he kept trying to get into the garage through the place where the former hole had been – yet the 2 month old cat had never been alive when there was a hole there! So – one might think it was a reincarnated cat using a former memory, except for what HPB says. So I take it when possible that actually the astral spooks of former cats innocently obsess, or meld with the young kittens, to explain the individual similarity between generations.
There are lots of accounts of animal ghosts, and many people have sensations of former pets being around after they have died.
Herbert Coryn in his “The New Way” prison newsletter of many years ago, gives an account similar to the above of transfer of memory or behavior. He had an old dog that would take one of his house slippers (he kept an old one around for the purpose) when it went to lay down near the stove, and rested his head on the house slipper like a pillow. The dog passed away after many years, and shortly thereafter a friend offered him a new pup. Coryn adopted the pup – and what do you know, the pup shortly adapted the same practice with the slipper.
Theosophy teaches that all forms of Life are on an evolutionary journey, each individualization – Human, animal, plant – is the expression of a god-spark or monad. While animals may not reincarnate in the same sense that humans do, still there must be some means of advance through learning experience or they wouldn’t actually be evoloving. Perhaps animals are more closely an expression of a species-specific unconscious or storehouse of experience – a cat or dog “collective unconscious.”
Theosophical philosophy holds that animals have the same higher principles that man has, but they are latent. I believe Blavatsky says elsewhere that the higher animals (apes, dogs perhaps, etc.) are near the point of developing a reincarnating ego, like humans, but not quite. They will enter the human kingdom in another Manvantara, except for some apes, which Purucker (and perhaps HPB somewhere) says will still enter the human kingdom during the next Round or active cycle of evolution on Earth. G. de Purucker gives a wider interpretation of reincarnation of animals in “The Esoteric Tradion.” 
 “The Esoteric Tradition,” vol. II, pp. 699-700: “Do animals reincarnate? The answer is yes. Animals reincarnate or reimbody themselves just as all other ‘animate’ entities do, humans included; for an animal has, or rather is, a ray from a reimbodying Monad equally with any other individualized and reimbodying entity, such as a human being. Yet there are certain important differences between human reincarnation and that of beasts. The human is a more or less highly individualized and awakened Ego, which the beast is not; for in the cases of the beasts the awakening egoity, otherwise the functioning of the manasic consciousness, is but in its elemental beginnings. The consequence of this is that human beings, or rather human egos, reincarnate as more or less individualized egos, each with an individual character, therefore with an individual ego-karman, possessing in consequence will-power or free choice, intellectual discrimination, judgment, and the moral instinct directing its definite choices to good or ill, all which faculties exist indeed in the beasts, but latent in them: they are but adumbrated or foreshadowed in the beast.
“Indeed, and going lower on the Ladder of Life to an even inferior Kingdom, that of the Plants: even vegetation reimbodies itself; and so also do the atoms in their own particular sphere. But in none of the Kingdoms below the Human are the individual cases of reimbodiment the reincarnation of more or less developed Ego-souls as is the case with individual human beings.” –