To produce a beneficial version of a trait in an animal, what type of process is used?

If by "beneficial" you mean "more suited to its environment," then the answer would be natural selection. Natural selection is the process by which, over time, the individuals of a species that have the traits most suited to their environment survive at the highest rate, leading to an increase in the proportion of individuals that possess those "beneficial" traits. A note here: natural selection is not something that is "used." Rather, it is a process that occurs naturally. This is why they say the environment "selects" for certain traits in a species. This same process can be harnessed by humans to breed a larger population of animals that possess a certain valuable trait that only a few members possess at the current point in time, but this process is called "artificial selection."