There are some magical works with animals, however I wouldn't give it a terms such as "beast magic". You do not have to define any form of magic with a specific term which has the word "magic" in it. Anyhow, you can call it however you want. But this can be confusing for other people who would like to answer your question.
Well, witches work with "Familiars".
The first and most common type of familiar is one of a pet.
It can be a dog, a cat, a bird, a lizard, a snake, ect.
These type of familiars will help you with your spell casting and can also give you energy if ''they choose to'' to help you when you need an extra boost.
He/she will be drawn to just you and usually snubs everyone else.
They will also join you when you are casting and lend you their energy.
There are also astral familiars. The astral familiar is more or less any number of creatures that come to you during a meditation and/or when seeking them, which obtains certain characteristics that are necessary to you in your task, be it magickal or just at this time in your life.
Lets not forget about power animals.
In the shamanic belief every thing is alive and carries with it power and wisdom. Power animals are an essential component of shamanic practice. They are the helping spirit which add to the power of the shaman and are essential for success in any venture undertaken by the shaman.
Shamans believe that everyone has power animals - animal spirits which reside with each individual adding to their power and protecting them from illness, acting similarly to a guardian angel. Each power animal that you have increases your power so that illnesses or negative energy cannot enter your body. The spirit also lends you the wisdom of its kind.
There are also few other working with animal in the magical practices.
However, I am not familiar with what you call "beast gates" or any similar term. I do think that this is something fluffy that made you misunderstand some stuff.