The general motivation, first is bodhicitta, that has to be there. Think,

“The purpose of my life and the purpose of liberating animals is not just to achieve happiness for myself, and it’s not even to achieve liberation from samsara, the blissful state of peace, for myself. The purpose is to free the numberless sentient beings—the numberless hell beings, numberless hungry ghosts, numberless animals, numberless human beings, numberless suras, numberless asuras and numberless intermediate state beings in the numberless universes—to free all of them from the oceans of samsara’s sufferings, by being free from delusion and karma. Then, next is to bring everyone to enlightenment, the total cessation of obscurations and completion of realizations, by myself alone.

“Even though there are numberless buddhas and bodhisattvas helping every sentient being, I have to repay the kindness of sentient beings. I should take full responsibility for their past, present and future happiness, including enlightenment, because everything I have received is due to the kindness of every hell being, every hungry ghost, every animal, every human being, every sura, every asura and every intermediate state being."

You can elaborate from here on their kindness; how much you want to express, you can elaborate.

“To do this perfectly, without the slightest mistake, I must achieve full enlightenment, therefore I am liberating these animals from the oceans of suffering of samsara and in particular from the most unbearable suffering of the lower realms.”

This is the general motivation. Of course, your motivation can always be elaborated according to the lamrim and according to whatever you want.
 
Now first, who to liberate animals for? When we save the lives of animals, first is His Holiness the Dalai Lama, who is the supreme leader of all the sentient beings, the numberless sentient beings of the six realms, including us, every sentient being, where all our happiness comes from.

Think, “May His Holiness have a stable life and may all his holy wishes succeed immediately.”

Also, dedicate for:

• All the other holy beings, who are here to benefit sentient beings.
• The teachings of the Buddha to last a long time.
• The Sangha, who preserve the Dharma and spread the Dharma.
• Sentient beings to have a long life and success in their Dharma practice.
• The benefactors, who serve the teachings of Buddha and the Sangha. For the benefactors to have a long life and also to have success in offering service to the teachings of the Buddha and to the Sangha.
• All the sentient beings who do good things for others.
• All the evil beings to meet the Dharma and understand and accept reincarnation and karma, and for them to abandon the cause of suffering, negative karma, and to practice holy Dharma, which is the cause of happiness, and to have a long life.

So this is the motivation. Of course, dedicate for the specific people who are sick or who have life obstacles, for them to have a long life.