The definition of Muktze (Regarding this matter) it is something unnecessary for Shabbat. Most opinions state that pets are in the same category as any other animal and therefore cannot be carried on Shabbat. However, some opinions permit carrying animals as long as they are pets.
To avoid causing suffering to the animal, it can be pushed, dragged, or even its legs moved (without lifting it), with the aim of moving it away from suffering.
There is an obligation (independent of Shabbat) to feed animals whose food depends on the person, even before the person's own food but not before the person's drink (Shulchan Aruch Alter Rebbe, 167:9)
One may feed one's animals on Shabbat, since their food depends on the person. Regarding a dog, even if it has no owner, it is permissible to feed it a little. (Shulchan Aruch Alter Rebbe Orach Chaim 324:7).
On Yom Tov, although some food preparation work is permitted, this applies to humans, not animals. Therefore, preparing food for animals is prohibited on Yom Tov if it requires performing any of the prohibited tasks involved in such preparation (Shulchan Aruch, Orach Chaim 512:3).
The following are some quotes from books that deal with the topic.
It is forbidden to carry domestic animals, wild animals or birds.
Shulchan Aruch, Orach Chaim 308:39
It is forbidden to carry domestic animals, wild animals, or birds because they are not useful on Shabbat when they are alive; even a bird that serves to make a child play when he cries is forbidden to carry. .
Shulchan Aruch of the Alter Rebbe, Orach Chaim 308:78
Birds and similar creatures, like people, those who waste time on trivial and meaningless things, often play with them, since their use is to carry them from one place to another; it can be done, and they are not Muktze.
Shulchan Shlomo, Seif Katan 74
All animals are Muktze, even those that children play with, except if they are exclusively pets.
Igrot Moshe, Orach Chaim 5, 22, Seif 21