Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
The meat of the animals, the questioner is asking about, is Halal and there is no reason that makes it unlawful. He did not mention in his question whether the animals eat impurities or not. If they eat impurities then Muslim scholars have different opinions based on the extent of the impurity the animals consumes and the period of custody that could make their meat pure if they are kept away from eating impurity.
If you are asking about that the animals that eat or destroy others' belongings, it does not affect the lawfulness of their meat.
As for the question on whether the owner of the animals is responsible for the loss they have made or not, the scholars prefer the opinion that if the animals eat or destroy other’s belongings at night the owner is responsible; if they do that during the day he is no longer responsible for that loss.
Ibn Qudamah wrote in his book al-Mughni: 'If animals destroy crops at night their owner is responsible for that loss and if they do that in the day the owner is no more responsible.'
Allah knows best.